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Algebraic , Approximation: AGuide to Past

and Current Solutions ~ Birkhauser

Jorge Bustamante

Algebraic Approximation: A Guide to Past

and Current Solutions

Jorge Bustamante Facultad de Ciencias Físico-Matemáticas Benemérita Universidad Autónoma de Puebla Apartado Postal 1152 Puebla, Pue. C.P. 72000 Mexico [email protected]

ISSN 1660-8046 e-ISSN 1660-8054 ISBN 978-3-0348-0193-5 e-ISBN 978-3-0348-0194-2 DOI 10.1007/978-3-0348-0194-2 Springer Basel Dordrecht Heidelberg London New York Library of Congress Control Number: 2011941776 Mathematics Subject Classification (2010): 41-02, 41-03, 41A25, 41A27, 41A36, 41A50, 41A52 © Springer Basel AG 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained. Printed on acid-free paper Springer Basel AG is part of Springer Science +Business Media (www.birkhauser-science.com)

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Some Notes on Trigonometric Approximation 1.1 Early years . . . . . . . . . . . . . . . . . . . . . 1.2 Direct and converse results: a motivation 1.3 Some asymptotic results . . . . . . . . . . . . 1.4 An abstract approach . . . . . . . . . . . . . .

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1 4 9 9

2 The End Points Effect 2.1 Two different problems . . . . . . . . . . . . . . . . . . . 2.2 Nikolskii’s discovery . . . . . . . . . . . . . . . . . . . . . 2.3 Problems connected with Nikolskii’s result . . . . . 2.4 Timan-type estimates . . . . . . . . . . . . . . . . . . . . 2.5 Estimates with higher-order moduli . . . . . . . . . . 2.6 Gopengauz-Teliakovskii-type estimates . . . . . . . . 2.7 Characterization of some classes of functions . . . 2.8 Simultaneous approximation . . . . . . . . . . . . . . . 2.9 Zamansky-type estimates . . . . . . . . . . . . . . . . . 2.10 Fuksman-Potapov solution to the second problem 2.11 Integral metrics . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Lp , 0 < p < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 The Whitney theorem . . . . . . . . . . . . . . . . . . . . 2.14 Other classes of functions . . . . . . . . . . . . . . . . .

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11 12 13 16 19 23 27 34 44 46 56 62 64 66

3 Looking for New Moduli 3.1 The works of Potapov . . . . . . . . . . . . . . . . . 3.2 Butzer and the method of Fourier transforms 3.3 The τ modulus of Ivanov . . . . . . . . . . . . . . 3.4 Ditzian-Totik moduli . . . . . . . . . . . . . . . . . 3.4.1 Direct and converse results . . . . . . . . 3.4.2 Approximation in weighted spaces . . .

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vi

Contents

3.4.3 3.4.4 3.4.5 3.5 Felten

Marchaud inequalities . . . . . Simultaneous approximation A Banach space approach . . modulus . . . . . . . . . . . . . .

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86 87 92 99

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101 103 104 105 107

5 Construction of Special Operators 5.1 Estimates in norm . . . . . . . . . . . . . . . . . . . 5.2 Timan-type estimates . . . . . . . . . . . . . . . . . 5.3 Gopengauz estimates . . . . . . . . . . . . . . . . . 5.4 Bernstein interpolation process . . . . . . . . . . 5.4.1 Bernstein first interpolation operators 5.4.2 Chebyshev polynomials of second type 5.4.3 General Bernstein operators . . . . . . . 5.4.4 Other modifications . . . . . . . . . . . . . 5.5 Integral operators . . . . . . . . . . . . . . . . . . . . 5.6 Simultaneous approximation . . . . . . . . . . . . 5.7 Estimation with constants . . . . . . . . . . . . . . 5.8 The boolean sums approach . . . . . . . . . . . . 5.9 Discrete operators . . . . . . . . . . . . . . . . . . .

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115 119 124 131 134 134 136 138 139 150 159 162 169

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205

4 Exact Estimates and Asymptotics 4.1 Asymptotics for Lip1 (M, [−1, 1] . . 4.2 Estimates for W r . . . . . . . . . . . . 4.3 Asymptotics for C r,w [−1, 1] . . . . . 4.4 Estimates for integrable functions 4.5 Pointwise asymptotics . . . . . . . . .

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Preface This book contains an exposition of several results related with direct and converse theorems in the theory of approximation by algebraic polynomials in a finite interval. In addition, we include some facts concerning trigonometric approximation that are necessary for motivation and comparisons. The selection of papers that we reference and discuss document some trends in polynomial approximation from the 1950s to the present day. The book does not pretend to be a text for graduate students. We only want to ease the task of understanding the evolution of ideas and to help people in finding the correct references for a specific result. An important feature of the book is to put together some different known solutions to problems in algebraic approximation that are not collected in text books. This explains the large number of references. Almost all of the material appears in historical order, but the concepts are separated into groups in order to present a fuller picture of the state of the art in a specific problem. Several topics related with algebraic approximation are not included here. For instance, we do not discuss approximation with constraints, because that would double the length of the book. On the other hand, we do present a few facts concerning approximation by positive linear operators. I hope that this survey will be helpful to students and researchers interested in approximation by algebraic polynomials. Any suggestions that would help to improve these notes would be welcomed by the author. Jorge Bustamante Gonz´alez Benem´erita Universidad Aut´ onoma de Puebla Mexico

vii

Chapter 1

Some Notes on Trigonometric Approximation 1.1 Early years Let us denote by (C[a, b] C[0, 2π]) the space of all real continuous (2π-periodic) functions f provided with the uniform norm   f  = sup | f (x) | x∈[a,b]

=

sup

| f (x) | .

x∈[0,2π]

By Pn (Tn ) we denote the family of all algebraic (trigonometric) polynomials of degree not greater than n. In 1885 Weierstrass published his famous theorem asserting that, every continuous (periodic) function on a compact interval is the limit in the uniform norm of a sequence of algebraic (trigonometric) polynomials. We shall mention that, almost at the same time, Runge showed that an arbitrary continuous function can be approximated by means of a rational function and that rational functions can be approximated by means of polynomials [318] and [319]. But he did not formulate the result explicitly. In a modern notation, Weierstrass’s theorem can be written as follows: for any f ∈ C[a, b], lim En (f, [a, b]) = 0,

(1.1)

En (f, [a, b]) = inf{ f − P  : P ∈ Pn }

(1.2)

n→∞

where is called the best approximation of f (by algebraic polynomials) of order n. For trigonometric approximation the best approximation is defined analogously. That is, if f ∈ C[0, 2π], then En∗ (f ) = inf f − T . T ∈Tn

J. Bustamante, Algebraic Approximation: A Guide to Past and Current Solutions, Frontiers in Mathematics, DOI 10.1007/978-3-0348-0194-2_1, © Springer Basel AG 2012

1

2

Chapter 1. Some Notes on Trigonometric Approximation

After Weierstrass, several different proofs of the same result appeared. Among others, there are some due to Lebesgue (1898, who used approximations of a function by broken lines [228]), Lerch (1892 and 1903, who approximated by a polygonal line and then by a Fourier series, [232] and [233]), Volterra (1987, who used ideas very similar to Lerch’s, [401]), Borel (1905, [36]), Landau (1908, who used a singular integral [224]), Simon (1918, who modified Landau’s ideas in order to approximate by finite sums [341]), de la Vall´ee-Poussin (1908, who provided an elegant proof in the trigonometrical case using a special integral, see [82] and [85]) and Kryloff (1908, who used a discrete version of de la Vall´ee-Poussin’s integral, [222]). For interesting comments related with these results see the paper of Pinkus [281]. As Jackson remarked [177], a time came when there was no longer any distinction in inventing a proof of Weierstrass’s theorem, unless the new method could be shown to possess some specific excellence. At that time it was known that there exist some connections between the smoothness of a function and its approximation by partial sums of Fourier series. These ideas can be found in Picard’s book [280]. It was Lebesgue in 1908 who formally stated the problem of studying the relation between smoothness and best approximation [229]. He considered the problem for Lipschitz functions. We say that L ∈ Lipα [a, b] (0 < α ≤ 1), if there exists a constant K = K(f ) such that | f (x) − f (y) | ≤ K | x − y |α .

(1.3)

We also set Lipα (M, [a, b]) = {f : [a, b] → R :| f (x) − f (y) | ≤ M | x − y |α }.

(1.4)

Lebesgue proved that if f ∈ Lip1 [a, b], then  En (f ) ≤ C (logn)/n. In [82] de la Vall´ee-Poussin improved the estimate by showing that √ En (f ) ≤ C/ n. The study of the special function g(x) =| x |, x ∈ [−1, 1], played an important role. It belongs to g ∈ Lip1 [−1, 1]. In 1908 de la Vall´ee-Poussin [81] constructed a polynomial Pn such that C | Pn (x)− | x | |< . n Two years later he proved that we can not find polynomials Pn satisfying | Pn (x)− | x | |
0, and r

ω(t + s) ≤ ω(s) + ω(t).

(1.7)

Theorem 1.2.1. Fix f ∈ C[0, 2π], r, s ∈ N0 and σ such that r < σ < s and let {Tn } be the sequence of polynomials of the best approximation for f . The following assertions are equivalent: (i)

En∗ (f ) = f − Tn  = O(n−σ ),

(ii)

ωs (f, t) = O(tσ ), (s) Tn 

(iii)

(n → ∞), (t → 0),

= O(n−(σ−s) ), (r) Tn 

(n → ∞),

(iv)

f ∈ C r [0, 2π] and f (r) −

(v)

f ∈ C [0, 2π] and ω1 (f

),

(0 < σ − r < 1),

(vi)

f ∈ C r [0, 2π] and ω2 (f (r) , t) = O(tσ−r ),

(0 < σ − r < 2).

r

(r)

= O(n−(σ−r) ),

, t) = O(t

σ−r

(n → ∞),

The fact that the assertions in Theorem 1.2.1 are equivalent not only in C[0, 2π] but in the setting of normed spaces was proved by Butzer and Scherer ([51] and [52]). They showed that essentially what we need is to have on hand appropriate Jackson and Bernstein-type inequalities. The abstract approach will be presented in the last section of this chapter. The assertion (v) ⇒ (i) was proved by Jackson [175] and [176] (the statement presented here is not the original). Different versions were later developed by Favard [115], Akhieser and Krein [1] and Korneichuk [207]. Theorem 1.2.2 (Direct result). For each r ∈ N, f ∈ C r [0, 2π] and n ∈ N0 ,  C(r) Kr (r) 1 En∗ (f ) ≤ ω f , and En∗ (f ) ≤ r+1 f (r) , n nr n

1.2. Direct and converse results: a motivation

5

where Kr is Favard’s constant defined by ∞ 4 (−1)k(r+1) . Kr = (2k + 1)r+1 π

(1.8)

k=0

It is known that K0 = 1, K1 = π/2, K2 = π 2 /8, K3 = π 3 /24 and π2 4 π = K2 < K4 < · · · < < · · · < K3 < K1 = . 8 π 2 In 1912 Bernstein studied the converse result. The proof of Bernstein is a model for almost all converse theorems and was based on Bernstein’s inequality Tn(r) ≤ nr Tn 

(Tn ∈ Tn ),

and n ∈ N) which follows from (1.5) by induction. Let W[0, 2π] be the class of all functions f ∈ C[0, 2π] for which there exists a constant C = C(f ) such that ω(f, t) ≤ C t(1+ | ln t |). Theorem 1.2.3 (Bernstein). Fix α ∈ (0, 1] and f ∈ C[0, 2π] and suppose there exists a constant C such that En∗ (f ) ≤ cn−α . Then, if α < 1, f ∈ Lipα [0, 2π] and if α = 1, f ∈ W[0, 2π]. In 1919 de la Vall´ee-Poussin [85] (following Bernstein’s ideas) proved the following theorem. Theorem 1.2.4. Let Ω : [a, ∞) (a > 1) be a decreasing function such that  ∞ Ω(u) lim Ω(t) = 0 and du < ∞. t→∞ u a If p ∈ N, f ∈ C[0, 2π] and En∗ (f ) ≤ Ω(n)n−p , then f (p) exists and     ∞ a/t Ω(u) du . Ω(u)du + ω(f (p) , t) ≤ C t u a 1/t

(1.9)

Following Freud, here O may not be substituted by o. Thus if, for some r ≥ 0 and 0 < α < 1, there is a polynomial Tn ∈ Tn such that for n ∈ N we have f − Tn  ≤

C , nr+α

then f ∈ C r [0, 2π] and f (r) ∈ Lipα [0, 2π]. That is (i) ⇒ (v) (for α = 1). Thus the problem of the characterization of the class of functions with an rth derivative in Lipα [0, 2π] was completely solved for the case 0 < α < 1. The case α = 1 is not included in the last results. Bernstein proved that condition En∗ (f ) = O(n−1 ) does ∞ −2 not imply f ∈ Lip1 [0, 2π]. In particular the function f (x) = sin(kx) k=1 k provides a counterexample.

6

Chapter 1. Some Notes on Trigonometric Approximation

Zygmund showed that we must consider a wider class: Z[0, 2π] = {f : C[0, 2π] → R : | Δ2h f (x) | ≤ Ah, h ∈ (0, 1] }. Theorem 1.2.5 (Zygmund, [420]). If f ∈ C[0, 2π], r ∈ N and 0 < α < 2, then En∗ (f ) = O(n−r−α ) if and only if | Δ2h f (r) (x) |≤ Chα . Thus the assertion (i) ⇔ (vi) for σ − r = 1 is proved. In [420] Zygmund also included some results related with non-periodic functions. Montel [253] studied several facts concerning the class Z. For 1 < p < ∞ Timan [382] proved a sharper version of the Jackson-type inequality for the best trigonometric approximation:  r 1/s

−r sr−1 ∗ s k Ek (f )p ≤ C(r, p)ωr (f, 1/n)p , (1.10) n k=1

where s = max{p, 2}. The converse inequality is given in the following form: Theorem 1.2.6 (Timan, [381]). For 1 < p < ∞, q = min{p, 2} and f ∈ Lp [0, 2π], one has  r 1/q

−r rq−1 ∗ q ωr (f, 1/n)p ≤ C(r, p)n k Ek (f )p . k=1

There is also an equivalent relation. Theorem 1.2.7 (Zygmund, [421]). For 1 < p < ∞, q = min{p, 2} and f ∈ Lp [0, 2π], one has  1/q 1/2 ωr+1 (f, u) r du . ωr (f, t)p ≤ C(r, p) t uqr+1 t This last relation is sometimes called a sharp Marchaud inequality. Some extensions were given by Ditzian [96] In 1949 Zamansky showed that (i) ⇒ (iii). Theorem 1.2.8 (Zamansky, [415]). Let ϕ be a positive strictly increasing or decreasing continuous function and fix f ∈ C[0, 2π] and m ∈ N. Suppose that for each n ∈ N there is a polynomial Tn ∈ Tn such that f − Tn  ≤ n1−m ϕ(n), then there are constants C1 , C2 and C3 such that Tn(m)  ≤ C1 + C2 nϕ(n) + C3



n

ϕ(u)du. 1 (m)

In particular if f − Tn  = O(n−β ) for β > 0, then Tn β < m.

 = O(nm−β ) with

1.2. Direct and converse results: a motivation

7

The assertion (i) ⇒ (iv) is due to Stechkin. Theorem 1.2.9 (Stechkin, [347]). Fix k ∈ N {Fn } be a non-increasing and let k−1 sequence of non-negative numbers such that ∞ n Fn < ∞. Let f ∈ C[0, 2π] n=1 and {Tn } (Tn ∈ T) a sequence such that f − Tn  ≤ Fn+1 ,

, n ∈ N.

Then f ∈ C k [0, 2π] and there exists a constant C such that ⎛ ⎞ ∞

j k−1 Fj ⎠ , f (k) − Tn(k)  ≤ C ⎝nk Fn+1 + j=n+1

for each n ∈ N. In 1967–1968 Butzer-Pawelke [46] and Sunouchi [360] proved the assertion (iii) ⇒ (i). In fact Butzer and Pawelke considered the problem for L2 [0, 2π] and Sunouchi for all Lp [0, 2π] spaces and C[0, 2π]. Theorem 1.2.10. Fix m ∈ N, β ∈ (0, m) and f ∈ C[0, 2π] and let {Tn } be the (m) sequence of polynomials of the best approximation for f . If Tn  = O(nm−β ), −β then f − Tn  = O(n ). The results recalled above also hold in Lp [0, 2π] spaces (1 ≤ p < ∞). For instance, for 0 < α < r, En (f )p = O(n−α ) ⇐⇒ ωk (f, t)p = O(tα ).

(1.11)

We can interpret the equivalence (1.11) in two different forms. a) The classical Lipschitz spaces are characterized in terms of the best trigonometric approximation. b) A class of functions with a given rate for the best trigonometric approximation is characterized in terms of Lipschitz classes. For trigonometric approximation both assertions are the same. In algebraic approximation the situation is different. Some results in algebraic approximation were obtained by reduction to the trigonometric case. If f ∈ C[−1, 1], with the change of variable x → cos θ we obtain an even 2π-periodic function g(θ) = f (cos θ). If Pn ∈ Pn is the polynomial of best approximation for f , then Tn (θ) = Pn (cos θ) is the trigonometric polynomial of best approximation for g. Therefore, if f ∈ C 1 [−1, 1], then En (f ) = En (g) ≤

π π g   ≤ f  , 2(n + 1) 2(n + 1)

where we have used the relations g  (cos θ) = f  (cos θ) sin θ and | sin θ |≤ 1. Thus the precision has been changed. In this way, some theorems related the approximation of non-periodic functions by algebraic polynomials were obtained. For

8

Chapter 1. Some Notes on Trigonometric Approximation

instance, for every f ∈ C[a, b] and n ∈ N there exists a polynomial Pn ∈ Pn such that, for all x ∈ [a, b], | f (x) − Pn (x) | ≤ C ω(f, (b − a)/n)

(1.12)

([179], p. 15) and, if f ∈ C r [a, b], then for each n ∈ N (n > r) there exists a polynomial Pn ∈ Pn such that, for all x ∈ [a, b], | f (x) − Pn (x) | ≤

 C(b − a)p (p) b − a ω f , , np n−r

([179], p. 18). In both cases the constant C does not depend upon f or n. The theory of best approximation of functions by algebraic polynomials was not as complete as in the trigonometric case. A characterization of the class Lipα [−1, 1] (0 < α ≤ 1) in terms of the best approximation was not known. We notice that (1.12) can be written in the more precise form  b−a . En (f ) ≤ 12ω f, 2n For the converse results Bernstein can not obtain an analogous form of Theorem 1.2.3. He only found properties on proper subintervals. Theorem 1.2.11 (Bernstein). Fix α ∈ (0, 1] and f ∈ C[a, b] and suppose there exists a constant C such that En (x) ≤ cn−α . Then for each couple of numbers c and d satisfying a < c < d < b one has f ∈ Lipα [c, d] (if α < 1) and f ∈ W[c, d] (if α = 1). The restriction to proper √ subintervals of [a, b] is essential. It was known that for the function f (x) = 1 − x2 one has En (f, [−1, 1]) < 2/(πn), while f ∈ / Lipα [−1, 1] for any α > 1/2. Thus a result like Theorem 1.2.4 holds only on proper subintervals of [a, b]. Some years later, in 1956, Csibi proved that we can obtain a conclusion for the full interval if the rate of approximation is faster. Theorem 1.2.12 (Csibi, [77]). Let Ω be as in Theorem 1.2.4. If p ∈ N, f ∈ C[a, b] and En (f, [a, b]) ≤ Ω(n2 )n−2p , then f (p) exists and ω(f (p) , t) satisfies (1.9). Bernstein [30] also considered the case of a higher rate of convergence of the best approximation. In particular, he proved that, if for each r ∈ N, lim nr En (f, [a, b]) = 0,

n→∞

then f has derivatives of all order in (a, b).

1.3. Some asymptotic results

9

1.3 Some asymptotic results There are two typical problems: to estimate the error for a fixed class of function (see (1.15)) and to consider the asymptotic of error in the class (see (1.16)). For the first problem, fix a bounded set M ⊂ C[0, 2π], n ∈ N and define En∗ (M ) = sup En∗ (f ).

(1.13)

f ∈M

Of course, we can obtain the exact value of En∗ (M ) only for some special sets M . Even more, it is not easy to know if the sup is attained in some element of M . The analysis of this problem was motivated by some results of Favard and Akhieser-Krein. Set W r+1 (M, [0, 2π]) = {f : f (r) ∈ A.C.[0, 2π], f (r+1) ∞ ≤ M }.

(1.14)

Favard [115] and Akhieser-Krein [1] proved that En∗ (W r (1, [0, 2π]) =

Kr , nr

(1.15)

where Kr is Favard’s constant (1.8), and Nikolskii noticed in [270] that there exist functions f ∈ W r (1, [0, 2π]) for which lim sup nr En∗ (f ) = Kr .

(1.16)

n→∞

1.4 An abstract approach Let us present an abstract approach introduced by Butzer and Scherer in [51], [52] and [53] (see also [185]). Let X be a normed space with norm  · X and {Mn } be a sequence of linear subspaces of X such that Mn ⊂ Mn+1 and ∪∞ n=1 Mn is dense in X. For f ∈ X, the best approximation of f by elements of Mn is defined as En (f ) = inf f − gX . g∈Mn

We assume that, for each f ∈ X and n ∈ N, there exists gn = gn (f ) ∈ Mn such that En (f ) = f − gn X . Let Y be a linear subspace of X with a seminorm | · |Y such that, for each n ∈ N, Mn ⊂ Y . Set K(f, t; X, Y ) = inf {f − gX + t | g |Y }, g∈Y

(t > 0, f ∈ X).

Theorem 1.4.1. Let X be a normed linear space and {Mn } be a sequence of linear subspaces of X such that Mn ⊂ Mn+1 and ∪∞ n=1 Mn is dense in X. Fix two linear

10

Chapter 1. Some Notes on Trigonometric Approximation

subspaces of X, Y and Z, with a seminorms | · |Y and | · |Z respectively such that, for each n ∈ N, Mn ⊂ Y and Mn ⊂ Z. Fix two real numbers ρ and σ (0 ≤ σ < ρ) and assume there exist constants A, B, C and D such that En (g) ≤

A | g |Y , nρ

| gn |Y ≤ B nρ gn ,

(g ∈ Y, gn ∈ Mn , n ∈ N), (1.17)

En (h) ≤

C | h |Z , nσ

| gn |Z ≤ D nσ gn 

(h ∈ Z, gn ∈ Mn , n ∈ N). (1.18)

If Z is a Banach space with the norm ·Z = ·X + | · |Z , then the following assertions are equivalent for f ∈ X and σ < s < ρ (i) (ii) (iii) (iv)

En (f ) = O(n−s ), as n → ∞. If En (f ) = f − gn X , then | gn |Y = O(nρ−s ), as n → ∞. If f ∈ Z and En (f ) = f − gn X , then | f − gn |Z = O(nσ−s ), as n → ∞. K(f, tρ , X, Y ) = O(ts ), as t → 0.

Chapter 2

The End Points Effect 2.1 Two different problems As we have noticed, the theorems related with direct and converse results for trigonometric approximation can not be translated word by word to the case of algebraic approximation. Thus we have two different questions: 1. Given a modulus of smoothness, how can the associated (generalized) Lipschitz classes be characterized with the help of approximation by means of algebraic polynomials? 2. How can the class of functions with a given rate of algebraic polynomial approximation (say En (f ) ≤ M/nα ) be characterized in terms of properties related with smoothness and/or differentiability? Some solutions to the first problem were given by Nikolskii, Timan and Dzyadyk. They considered the space of continuous functions and uniform approximation. In this approach the use of the quantity √ 1 − x2 1 + 2 Δn (x) = n n

(2.1)

was essential. Fuksman presented the first results related with solutions of the second problem. Fuksman obtained a characterization of functions f ∈ C[−1, 1] for which En (f ) ≤ M/nα (0 < α < 1) with the help of a local modulus of continuity. After the works of Timan some different problems were considered: (i) When can Δn (x) be changed by √ 1 − x2 ? δn (x) = n

J. Bustamante, Algebraic Approximation: A Guide to Past and Current Solutions, Frontiers in Mathematics, DOI 10.1007/978-3-0348-0194-2_2, © Springer Basel AG 2012

(2.2)

11

12

Chapter 2. The End Points Effect

(ii) When can the polynomials used in approximating functions also approximate the derivatives? (iii) When can the estimates given in terms of the first modulus of continuity be improved by using higher-order moduli? (iv) Is it possible to obtain estimates which combine (i) and (ii) (or other relations)?

2.2 Nikolskii’s discovery Since the known estimates for trigonometric approximation did not always lead to optimal results in algebraic approximation, some new methods were needed. In 1946 Nikolskii made an important advance by considering point-wise estimation by means of a sequence of linear operators. Let {Tn } be the sequence of Chebyshev polynomials Tn (x) = cos(n arccos x). It is known that these polynomials are orthogonal with respect to the measure √ 2dt/π 1 − t2 on the interval [−1, 1]. As usual, for f ∈ C[−1, 1] the Fourier-Chebyshev coefficients are defined by 2 an (f ) = π



1

−1

f (t)Tn (t)dt √ . 1 − t2

For f ∈ C[−1, 1] and x ∈ [−1, 1] define a0 (f )

+ kλn,k ak (f ) Tk (x) 2 n

Un (f, x) =

k=1

where

kπ π . cot 2n 2n Theorem 2.2.1 (Nikolskii, [271]). For each n ∈ N, one has Un : C[−1, 1] → Pn . Moreover, if f ∈ Lip1 (M, [−1, 1]) (see (1.4)) and n ∈ N, then √  M π 1 − x2 log n +|x|O (2.3) | f (x) − Un (f, x) |≤ 2 n+1 n2 λn,k =

and O can not be replaced by o. Proof. It is sufficient to consider the case M = 1. Notice that every function in Lip1 [−1, 1] is absolutely continuous. Let us write    ∞ n−1 



sin kt 2 π   , In = λn,k sin(kt) dt K(t) = K(t) −  t π 0  k=1

k=1

2.3. Problems connected with Nikolskii’s result

and 2 Jn = π



π 0

13

  n−1  

  λn,k sin(kt) sin t dt. K(t) −   k=1

By setting x = cos θ and integrating by parts, one has  1 2π a0 + K(t) sin(t + θ)f  (cos(t + θ))dt f (cos θ) = 2 π 0 and 1 a0 + Un (f, cos θ) = 2 π



2π 0

n−1

 λn,k sin(k + t) sin(t + θ)f  (cos(t + θ))dt.

k=1



Since | f (cos(t + θ)) |≤ 1,

  n−1  

  sin(k + t) | sin(t + θ) | dt. K(t) −   0 k=1  ≤ In | sin θ | +Jn | cos θ |≤ In 1 − x2 + Jn | x | .

1 | f (cos θ) − Un (f, cos θ) | ≤ π



2π

Then Nikolskii proved that In = π/(2n) and Jn = O(ln n/n2 ).



There is a great difference with Jackson’s theorem: the position of x on the √ interval [−1, 1] is taken into account in the factor 1 − x2 . Timan and Dzyadik [380] proved that, if f ∈ C r [a, b] and f (r) is quasi-smooth (Zygmund), then En (f ) = O(n−(r+1) ). The sentence improves a result of Montel for En (f ) which gave an estimate only inside of the interval.

2.3 Problems connected with Nikolskii’s result Nikolskii’s result motivated several investigations on the possibility of approximation (including the asymptotically best approximation) of functions of various classes by algebraic polynomials and many results concerning the improvement of approximation at the endpoints of the segment [−1, 1]. In 1958 Lebed [226] gave an extension of Nikolskii’s theorem by considering functions in the Zygmund class: Z[−1, 1] = {f : C[−1, 1] → R : | Δ2h f (x) | ≤ M h, h ∈ (0, 1] }. Theorem 2.3.1 (Lebed, [226]). If f ∈ C m [−1, 1] and f (m) ∈ Z[−1, 1] (with constant M ), then there exists a sequence {Pn } (Pn ∈ Pn ) such that  log n m Δn (x) + 2 , | f (x) − Pn (x) | ≤ C(m) M (Δn (x)) n where Δn (x) =

  1 − x2 + | x | /n .

(2.4)

14

Chapter 2. The End Points Effect

The factor π/2 in (2.3) cannot be replaced by a smaller one (see (4.2)). Temlyakov proved the existence of a sequence {Pn } for which an estimate with specific constants in both terms holds. His construction was not obtained by means of a sequence of linear operators, but he strengthened inequality (2.3) by omitting log n in the remainder. Theorem 2.3.2 (Temlyakov, [372]). Assume f ∈ Lip1 (1, [−1, 1]). For any natural number n there exists an algebraic polynomial Pn of degree n such that √ π2 | x | π 1 − x2 + | f (x) − Pn (x) |≤ . (2.5) 8n2 2n Proof. The proof is based on an inequality for the best trigonometric approximation of a differentiable function: En (h) ≤

Kr En (h(r) ), (n + 1)r

r ∈ N,

(2.6)

where Kr is the Favard constant. Since K1 = π/2, what we need is a good representation of the function g(t) = f (cos t). If ∞

a0

+ −f (cos t) = ak cos(kt) 2 

k=1

and

ϕ(t) =

∞

ak k=1

k

sin(kt),

then g can be written as g(t) = − where

 σ(t) =

a0 cos t + ϕ(t) sin t + σ(t) cos t + G(t), 2 t

ϕ(s)ds 0

and

G (t) = −σ(t) sin t.

Now, let un−1 and vn−1 be the trigonometric polynomial of best approximation of order n−1 of the functions ϕ and σ respectively and define Pn (cos t) = un−1 (t) sin t and Qn (cos t) = −vn−1 (t) cos t. Since E0 (ϕ ) ≤ 1, it follows from (2.6) that | ϕ(t) − Pn (cos t) | ≤| ϕ(t) − un−1 (t) || sin t |≤ En−1 (ϕ) | sin t | π π | sin t | ≤ En−1 (ϕ ) | sin t |≤ 2n 2n and | σ(t) cos t − Qn (cos t) | ≤| σ(t) − vn−1 (t) || cos t |≤ En−1 (σ) | cos t | K2 K2 ≤ 2 En−1 (ϕ ) | cos t |≤ 2 | cos t | . n n Finally, since d −G (arccos x) −G (t) √ = σ(t) = σ(arccos x), G(arccos x) = = sin t dx 1 − x2

2.3. Problems connected with Nikolskii’s result

15

there exists Sn ∈ Pn such that | (G(arccos x)) − Sn (x) |≤ En (σ) ≤ Thus, if we define

 Rn (x) = G(π/2) +

then

K2 K2 En (ϕ ) ≤ . (n + 1)2 (n + 1)2

0

x

Sn (y)dy

 x    K2 | x |   ((G(arccos y)) − Sn (y))dy  ≤ . | G(arccos x) − Rn (x) |=  (n + 1)2 0 The proof finishes by considering the polynomial P (x) = −a0 x/2 + Pn (x) + Qn (x) + Rn (x).



The second term in (2.5) was written as O(| x | /n2 ) in the original statement, but we prefer to present here what was really proved. If we want to compare (2.5) with (2.3), for f ∈ Lip1 (M, [−1, 1]) the last term in (2.5) should be multiplied by M . It could be avoided, if such a term can be replaced by zero. But Temlyakov did not know whether such a term can be removed. However, in the same paper he proved the following assertion. For each natural number n one can find a function fn ∈ Lip1 (1, [−1, 1] for which there exists no polynomial Pn ∈ Pn such that √ π 1 − x2 . | fn (x) − Pn (x) |≤ 2(n + 1) From Theorem 2.3.2, making use of some arguments of Teliakovskii [371], Temliakov obtained the following theorem (the constant in O(1/n) was not given). Theorem 2.3.3 (Temlyakov, [372]). Let f ∈ Lip1 (1, [−1, 1]). For any natural number n there exists an algebraic polynomial Pn of degree n such that √ π 1 − x2 (1 + O(1/n)) . | f (x) − Pn (x) |≤ 2(n + 1) In the chapter devoted to asymptotics we will include some other results. For differentiable functions Ligun presented in 1980 a version which provides some information concerning the constants. Theorem 2.3.4 (Ligun, [236]). Let r be an odd number. Then for any function f ∈ C r [−1, 1] there exists a sequence of algebraic polynomials {Qn,r (x)} of degree not greater than n ≥ r such that, uniformly with respect to x ∈ [−1, 1],    Kr (δn (x))r  (r) 1 (r) 1 ω f , πδn (x) + O ω f , . | f (x) − Qn−1,r (x) |≤ 2 nr n

16

Chapter 2. The End Points Effect

The proof of the last theorem is very long and technical, so it will not be included here. But we notice that the construction was obtained by means of linear operators. For a given modulus of continuity ω and r ∈ N, the associated Lipschitz class is defined by Hωk = {f ∈ C[a, b] : ωk (f, t) ≤ C(f )ω(t)} . Theorem 2.3.5 (Polovina, [286]). Let w be a modulus of continuity. For each function f ∈ C 1 [−1, 1] such that f  ∈ Hw , there exists a sequence of polynomials {Pn (f, x)} (Pn ∈ Pn ) such that 1 | f (x) − Pn−1 (x) | ≤ 2



√ π 1−x2 /n



w(t)dt + o 0

1 ω n

 1 . n

Moreover, if w is a concave modulus of continuity, 1/2 can be changed to 1/4. The constant 1/4 cannot be made smaller. In [18] Bashmakova presented a similar result for functions f such that f  ∈ Hw , for a continuous and concave modulus of continuity.

2.4 Timan-type estimates In [373] Timan proved that, if f ∈ Lipα (M, [−1, 1]) and Sn (f, x) is the nth partial sum of the Fourier-Chebyshev series of f , then 2α+1 M (1 − x2 )α/2 log n | f (x) − Sn (f, x) |≤ n2 π2





π/2

t sin tdt + O α

0

1 nα

.

Later, in 1951, he [374] improved the Nikolskii estimate as follows: for f ∈ Lipα ([−1, 1]) (0 < α ≤ 1) one can find a sequence {Pn } (Pn ∈ Pn ) such that | f (x) − Pn (x) |≤

C nα

  α |x| ( 1 − x2 )α + . n

(2.7)

In the same year he generalized this result. For the proof we need the Jackson (also called Jackson-Matsuoka) kernels. [250] 

2s sin(nt/2) , sin(t/2) π is chosen from the condition π −1 −π Kn,s (t)dt = 1. Kn,s (t) = cn,s

where cn,s

(2.8)

2.4. Timan-type estimates

17

Theorem 2.4.1 (Timan, [375]). For r ∈ N0 there exists a constant Cr such that, for each f ∈ C r [−1, 1] and n ∈ N, one can find a polynomial Pn (f ) ∈ Pn satisfying √ r   √ 1 − x2 1 1 1 − x2 (r) (2.9) ω f , | f (x) − Pn (f, x) |≤ Cr + 2 + 2 . n n n n Proof. The proof presented in [379] (p. 262–266) follows by an inductive argument with respect to r. Here we show the case r = 0. Define  1 π Q2n−2 (f, x) = f (cos t) [Kn,2 (t + y) + Kn,2 (t − y)]dt, π 0 where x = cos y. It can be proved that Q2n−2 (f ) ∈ P2n−2 . On the other hand,   π  1   | f (x) − Q2n−2 (f, x) |=  [f (cos y) − f (cos t)] [Kn,2 (t + y) + Kn,2 (t − y)]dt π 0  1 π ≤ ω(f, | cos y − cos t |) [Kn,2 (t + y) + Kn,2 (t − y)]dt π 0    π   t+y 1 t − y  [Kn,2 (t + y) + Kn,2 (t − y)]dt. = ω f, 2 sin sin 2π −π 2 2  Let us estimate the integral corresponding to Kn,2 (t + y) (the other one can be estimated with similar arguments).    π   1 t+y t − y   Kn,2 (t + y)dt ω f, 2 sin sin 2π −π 2 2   1 π/2 = ω (f, 2 |sin(t) sin(t + y)|) Kn,2 (2t)dt π −π/2  1 π/2 [ω(f, t2 ) + ω(f, t | sin y |)] Kn,2 (2t)dt. ≤ π 0 If we consider that ω(f, t2 ) ≤ (1 + n2 t2 ) ω(f, 1/n2 ) and ω(f, t | sin y |)) ≤ (1 + nt) ω(f,



1 − x2 /n),

it is sufficient to verify that there exists a constant C (independent of n) such that  π/2 [1 + nt + (nt)2 ] Kn,2 (2t)dt ≤ C.  0

In particular, if f (r) ≤ M , | f (x) − Pn (x) |≤

M Cr nr



1 − x2 +

|x| n

r .

(2.10)

18

Chapter 2. The End Points Effect

Theorem 2.4.2 (Timan [376]). If w is a modulus of continuity for which  ∞

1 1 w 0 for t > 0 and f (0) = f (1). It can be proved that there is a constant A such that  1 1 . ≤ Aω2 f, n2 n 2) Set g(x) = f (1 − x), x ∈ [0, 1]. It can be proved that there are extensions F and G of the functions f and g to the interval [0, 4] such that ω2 (F, t) ≤ 25ω2 (f, t)

and

ω2 (G, t) ≤ 25ω2 (g, t).

3) Define 

1/3

ϕ(x) = 2 0

and

 ψ(x) = 2 0

1/3

F (x2 + 9u2 )Dn,3 (u)du 9 F (x2 + u2 )Dn,3 (u)du. 2

Since | f (x2 ) − 2ψ(x) + ϕ(x) |   1   = f (x2 ) Dn,3 (u)du  1/3    1/3    2  9 2  2 2 2 F (x ) − 2F x + u + F x + 9u Dn,3 (u)du +  2 0  1/3  C1 9 ≤ 5 f  + 2 ω2 f, u2 Kn,3 (u)du n 2 0   1/3  2  1 9 1 C1 1 + (nu)2 Kn,3 (u)du ≤ C2 ω2 f, 2 , ≤ 5 F  + 2ω2 f, 2 n n 2 n 0 it is sufficient to approximate ϕ and ψ with polynomials in x2 . 4) Set     u+x u−x + Kn,3 du F (u2 ) Kn,3 3 3 −2   1 (2+x)/3 1 (2−x)/3 2 = F ((3u − x) )Kn,3 (u)du + F ((3u + x)2 )Kn,3 (u)du 2 (−2+x)/3 2 (−2−x)/3

P1 (x2 ) =

1 6



2

2.5. Estimates with higher-order moduli

1 = 2



1/3 −1/3

1 + 2



21

[F (x2 − 6xu + 9u2 ) + F (x2 + 6xu + 9u2 )Kn,3 (u)du 

−1/3

(2+x)/3

+ (−2+x)/3

1/3

 F ((3u − x)2 )Kn,3 (u)du

  (2−x)/3  −1/3 1 F ((3u − x)2 )Kn,3 (u)du + + 2 (−2−x)/3 1/3  1 1/3 = [F (x2 − 6xu + 9u2 ) + F (x2 + 6xu + 9u2 )Kn,3 (u)du + F O(n−5 ). 2 −1/3 Therefore | ϕ(x) − P1 (x2 ) | ≤

 0

1/3

ω2 (F, 6xu)Kn,3 (u)du + C3 F 

1 n4

 x  1/3 1 ≤ ω2 F, (1 + 6un)2 Kn,3 (u)du + C3 F  4 n 0 n    1 x . + ω2 F, 2 ≤ C4 ω2 F, n n 

The analogous construction for ψ is obtained by setting √  2     √ u+x √ u−x 2 2 + Kn,3 2 du. F (u2 ) Kn,3 P2 (x ) = 6 −2 3 3 2

Thus, if we define Pn (f, x2 ) = 2P2 (x2 ) − P1 (x2 ), then

  √  x 1 + ω2 F, 2 | f (x) − Pn (f, x) |≤ C5 ω2 F, n n

for x ∈ [0, 1]. If we realize the analogous construction for the function G, then | f (x) − Pn (G, 1 − x) | =| G(1 − x) − Pn (G, 1 − x) |   √  1 1−x + ω2 F, 2 . ≤ C5 ω2 F, n n For the final construction take m = [n/3] and define Pm (x2 ) = (1 − x)Pm (F, x) + xPm (G, x).

22

Chapter 2. The End Points Effect

We know that Pm ∈ Pn and | f (x) − Pn (x) | ≤ (1 − x) | f (x) − Pn (f, 1 − x) | +x | f (x) − Pn (G, 1 − x) |   √  √  x 1−x 1 + xω2 F, + ω2 F, 2 ≤ C6 (1 − x)ω2 F, n n n       x(1 − x) 1 + ω2 F, 2 .  ≤ C7 ω2 F, n n Dzyadyk also presented the following result. Theorem 2.5.2 (Dzyakyk, [109]). Assume that f : [−1, 1] → R and r ∈ N0 . One has f ∈ C r [−1, 1] and f (r) ∈ Lip1 [−1, 1] if and only if there exists a sequence of polynomials {Pn } (Pn ∈ Pn ), such that  r+1  Δn (x) . | f (x) − Pn (x) |= o n In 1959 Freud [123] (independently of Dzyadyk) constructed another sequence of polynomials for which a Timan result holds in terms of the secondorder modulus. Freud used the method of intermediate spaces. That is, he first approximates the function f by an adequate piecewise linear function g and then approximates g by polynomials. Freud said that the construction of polynomial kernels (such as the one used by Timan) is not a simple task and he stated the problem of obtaining similar results using differences of higher order and good estimations for the constants. The extension to moduli of smoothness of arbitrary order was given by Brudnyi in 1963. Theorem 2.5.3 (Brudnyi, [38]). Given r ∈ N, there exists a constant Cr such that, for each f ∈ C[−1, 1] and n ∈ N (n ≥ r − 1), there exists a polynomial Pn (f ) ∈ Pn such that | f (x) − Pn (f, x) |≤ Cr ωr (f, Δn (x)) . (2.14) Let Φk denote the class of all non-decreasing continuous functions ϕ such that, ϕ(0) = 0 and ϕ(t)/tk is non-increasing. Sometimes this last condition is changed by the weaker one: ϕ(t)/tk ≤ Cϕ(s)/sk , for 0 < s < t. Functions of these classes are said to be of the type of the kth order modulus of continuity. It is known that if ωk (f, t) = 0, then ωk (f, t) ∈ Φk (see [247] and [347]). For ϕ ∈ Φk and fixed constant M , set Hkϕ (M, [−1, 1]) = {f : [−1, 1] → R : ωk (f, h) ≤ M ϕ(h), h ∈ (0, 1/k]} and

W r Hkϕ (M, [−1, 1]) = {f ; f (k) ∈ Hkϕ (M, [−1, 1])}

(W 0 (M, [−1, 1]) = Hkϕ (M, [−1, 1])). Moreover set  W r Hkϕ (M, [−1, 1]). W r Hk [ϕ] = M>0

2.6. Gopengauz-Teliakovskii-type estimates

23

Theorem 2.5.4. Let k and r be natural numbers and assume  b−a ωk+r (f, u) du < ∞. ur+1 0 Then, for 0 < t ≤ b − a, ωk (f (r) , t) ≤ C(r, k)

 0

t

ωk+r (f, u) du. ur+1

This theorem was proved by Marchaud in [247] for k = 1. For k ≥ 2 the result was also proved by Brudnyi and Gopengauz in [41]. Guseinov [155] considered the problem of obtaining necessary and sufficient conditions on ϕ ∈ Φk and w ∈ Φk+r under which the equality Hwk+r = W r Hϕk holds. Theorem 2.5.5 (Guseinov, [155]). Let k, r be natural numbers and w ∈ Φr and set ϕ(t) = w(t)/tr . Then Hwk+r = W r Hϕk if and only if  t ω(u) w(t) du ≤ C(r, k) r . r+1 u t 0 Theorem 2.5.6 (Guseinov, [155]). Let k, r be natural numbers, w ∈ Φr+k and set ϕ(t) ∈ Φk . Then Hwk+r = W r Hϕk if and only if ϕ(t) ≤ Cw(t)/tr and  t ω(u) du ≤ Cϕ(t). r+1 u 0 For the classes Φk the Brudnyi theorem yields Theorem 2.5.7. If ϕ ∈ Φk , r ∈ N0 and k ∈ N, there exists a constant C = (r, k) such that, if f ∈ W r Hkϕ (1, [−1, 1]), then for any natural number n ≥ r + k − 1 there is an algebraic polynomial Pn of degree not greater than n, such that r

| f (x) − Pn (x) |≤ C(r, k) (Δn (x)) ϕ(Δn (x)).

(2.15)

2.6 Gopengauz-Teliakovskii-type estimates √ In a Timan-type estimate the term Δn (x) = 1 − x2 /n + 1/n2 appears. It is natural to ask whether such a term can be replaced by the simpler one δn (x) = √ 1 − x2 /n. A positive answer follows from the works of Teliakovskii (1966) using the first modulus of continuity Theorem 2.6.1 (Teliakovskii, [371]). Let r be a non-negative integer. There exists a constant Cr such that, for each f ∈ C r [−1, 1] and n > r one can find a polynomial Qn (f ) ∈ Pn such that   r | f (x) − Qn (f, x) |≤ Cr (δn (x)) ω f (r) , δn (x) . (2.16)

24

Chapter 2. The End Points Effect

Proof. We will consider only the case r > 0. Fix polynomials Pn (f ) such that the estimates (2.9) in Timan’s Theorem 2.4.1 holds and define Qn (x) = Pn (f, x) + Rn (f, x) where Rn (f ) ∈ Pq is the polynomial which interpolates the function f (x)−Pn (f, x) and its derivatives up to the order q = [r/2] at the points ±1. Teliakovskii proved that Qn satisfies (2.16). The proof also uses some ideas of simultaneous approximation which will be presented in another section. In particular it follows from Theorem 2.8.10 that  R 1 | f (k) (±1) − Pn (f )(k) (±1) |≤ 2(r−k) ω f (r) , 2 , n n where the constant R is independent of f and n. Now using the formula of interpolation of Hermite one has  q

(1 − x2 )j (r) 1 | Rn (f, x) |≤ R (2.17) ω f , n2 n2(r−j) j=0 and for 0 ≤ k ≤ r and x ∈ [−1, 1], | Rn(k) (f, x) |≤

C n2(r−q)

 1 ω f (r) , 2 . n

(2.18)

√ 1 − x2 , then from (2.9) and (2.17) one has √ r   √ 1 − x2 1 1 1 − x2 (r) + 2 + 2 ω f , | f (x) − Qn (x) | ≤ Cr n n n n r  √ q 1 1 − x2 1

√ +R ω f (r) , 2 n n (n 1 − x2 )r−2k j=0   r ≤ C (δn (x)) ω f (r) , δn (x) .

If 1/n ≤

√ Now assume 1/n ≥ 1 − x2 and suppose x > 0. If r > 0 (q + 1 ≤ r), then it follows from Theorem 2.8.10 and (2.18) that | f (x) − Qn (x) |    1 1  1     = (−1)q+1 ··· [f (q+1) (u) − Pn(q+1) (f, u) − Rn(q+1) (f, u)]duduq · · · du1    x u1 uq  1  1   1 R 1 ≤ 2(r−q−1) ω f (r) , 2 ··· duduq · · · du1 n n x u1 uq  R 1 ≤ 2(r−q−1) ω f (r) , 2 (1 − x2 )q+1 . n n

2.6. Gopengauz-Teliakovskii-type estimates

Since 1/n ≥

25

√ 1 − x2 ,   √  1 − x2 2 (r) 1 (r) , ω f , ω f , 2 ≤ √ n n n 1 − x2

we obtain

 1 − x2 | f (x) − Qn (x) | ≤ 2r−2q−1 (1 − x ) ω f , n n r   √ √  1 − x2 1 − x2 (n 1 − x2 )2q+1−r =R ω f (r) , n n r   √ √ 1 − x2 1 − x2 (r) . ≤R ω f , n n R



2 q+1/2

√

(r)

For x < 0 the result follows analogously.



In 1967 Gopengauz obtained a similar result in terms of the modulus of continuity of second order. Theorem 2.6.2 (Gopengauz, [151]). For each f ∈ C[−1, 1] and n ≥ 2, there exists Pn ∈ Pn such that | f (x) − Pn (x) | ≤ Cω2 (f, δn (x)) , where the constant C does not depend on f or n. Bashmakova and Malozemov gave an estimate including interpolation. Theorem 2.6.3 (Bashmakova and Malozemov, [17]). For f ∈ C[−1, 1] and −1 = x0 < x1 < · · · < xn = 1 (n > 1) there exists Pn (f ) ∈ Pn such that | f (x) − Pn (f, x) |≤ Aω (f, δn (x)) and, for x ∈ [−1, 1] and k = 1, . . . , m − 1,    | f (x) − Pn (f, x) |≤ Aω f, | x − xk |, | x − xk |/n . There is also a more complicated version of Theorem 2.6.2 for fractional derivatives. Theorem 2.6.4 (Shalashova, [336]). If f ∈ C[0, 1] has continuous derivatives of fractional order r (r = r + α, with r integer and α ∈ (0, 1)), there there exists for any n ≥ r − 1 a polynomial Pn ∈ Pn such that   r   x(1 − x) x(1 − x) , ω f (r) , | f (x) − Pn (x) − Axr |≤ Cr n n where Cr does not depend on f or n and A depends on f and r. For fractional r, the term Axr can not be omitted.

26

Chapter 2. The End Points Effect

A more general version of Theorem 2.6.1 was given by Stens in 1980. He proved the following theorem. The case α = 0 and α = σ are very illustrative. Theorem 2.6.5 (Stens, [348], [349]). Let s ∈ N and 0 ≤ α ≤ σ < s. For f ∈ C[−1, 1] the following assertions are equivalent: (i) There exists a sequence {Pn }, Pn ∈ Pn such that, for x ∈ [−1, 1],   √ ( 1 − x2 )σ−α . | f (x) − Pn (x) | ≤ C nσ (ii) For ϕ(x) =

√

1 − x2 , sup ||ϕα Δsh f ||C[−1,1] ≤ Ctσ .

|h|≤t

In [150] Gopengauz analyzed the following questions. Is it possible to obtain a Timan-type estimate but improving the rate of approximation in a certain interior point? Is it possible to obtain a speed of approximating at the ends of the segment greater than the one in Timan’s theorem? He considered that the rate of approximation on the whole segment is retained. Both questions were answered in the negative. For instance, he proved that an estimation of the form  ψ1 (| x − a |) + ψ2 (1/n) | f (x) − pn (x) | ≤ Cω f, , n is not possible for all f ∈ C[−1, 1], where | a |< 1 and ψi is an increasing function satisfying ψi (t) → 0 as t → 0 (i = 1, 2). Moreover, we can not replace the expression Δn (x) by o(Δn (x)). In 1985 Yu showed some inequalities which are not possible. In particular, the following is proved. Theorem 2.6.6 (Yu, [410]). Let r ∈ N∪{0} and C > 0. Then there exists a function f ∈ C r [−1, 1] such that there exists no polynomial Pn ∈ Πn satisfying   |f (x) − Pn (x)| ≤ C( 1 − x2 /n)r ωr+3 (f (r) , 1 − x2 /n). √ An analogous for the case in which the quantity 1 − x2 /n √ result is stated is replaced by 1 − x2 /n + n /n2 , n a positive number null sequence. In 2000 Gonska, Leviatan, Shevchuk and Wenz presented a result in the following form. Theorem 2.6.7 (Gonska, Leviatan, Shevchuk and Wenz, [148]). Let k ≤ r + 2 and assume that f ∈ C r [−1, 1]. Then there is a polynomial p ∈ Π2[(r+k+1)/2]−1 for which   | f (x) − p(x) | ≤ Cr ( 1 − x2 )r ωk (f (r) , 1 − x2 ), −1 ≤ x ≤ 1, (2.19)

2.7. Characterization of some classes of functions

27

where Cr depends only on r. Moreover, for each f ∈ C r [−1, 1] and n ≥ 2[(r + k + 1)/2] − 1, there is a polynomial Pn (f ) ∈ Pn , such that   (2.20) | f (x) − Pn (f, x) |≤ C(r) (δn (x))r ωk f (r) , δn (x) holds with a constant Cr depending only on r. It should be noted that it is impossible to replace 2[(r + k + 1)/2] − 1 by any lower figure. It has been shown by Yu [410] (see also Li [235]), that (2.20) is not valid if k > r + 2: assume that k > r + 2 ≥ 2, then for each n and every constant A > 0, there exists a function f = fr,k,n,A ∈ C r [−1, 1] such that, for any polynomial pn ∈ Pn , there is a point x ∈ [−1, 1] for which | f (x) − pn (x) |> A(δn (x))r ωk (f (r) , δn (x)) holds. One also has Theorem 2.6.8 ([148]). Given r ≥ 0, there exists a function f ∈ C r [−1, 1] such that, for any algebraic polynomial p, | f (x) − p(x) | √ lim sup √ = ∞. 2 x→−1 ( 1 − x )r ωr+3 (f (r) , 1 − x2 ) It is possible to construct a function which exhibits this phenomenon at both endpoints.

2.7 Characterization of some classes of functions As we recall, if f ∈ C[0, 2π], r ∈ N and α ∈ (0, 1), then f ∈ C r [0, 2π] and f (r) ∈ Lipα if and only if En (f )∗ = O(n−(r+α) ). The same result is not true in the non-periodical case. Some characterizations appeared in works of Timan and Dzyadyk. Theorem 2.7.1. Let f ∈ C[−1, 1], r a positive integer and α ∈ (0, 1). The following assertions are equivalent: i) f ∈ C r [−1, 1] and for each x ∈ [−1, 1], sup {h: |h|≤δ, |x+h|≤1}

| f (r) (x) − f (r) (x + h) |≤ C δ,

where C is a positive constant which does not depend on x or δ; ii) For each n ∈ N there exists Pn ∈ Pn such that, for each x ∈ [−1, 1],  r+α D 1 2 1−x + , | f (x) − Pn (x) | ≤ r+α n n where D is a positive constant which does not depend on x or δ.

28

Chapter 2. The End Points Effect

In 1957 Timan presented a converse result assuming that an estimate in terms of the function Δn (x) is known. For the definition of modulus of continuity see (1.7). Theorem 2.7.2 (Timan, [376]). Let ω be a modulus of continuity and f : [−1, 1] → R be a function and suppose there exists a sequence {Pn } (Pn ∈ Pn ) such that   |x| 1 , x ∈ [−1, 1]. 1 − x2 + | f (x) − Pn (x) | ≤ ω n n Then



1

ω(f, t) ≤ C t

ω(s) ds, s2

t

0 a/(1 + b) the modulus of continuity ω(fa,b )(t) = ta/(1+b) , whereas En (f ) ∼ n−2a/(2b+1) . Theorems 2.7.11 and 2.7.12 show, in particular, that the assertion of the inverse theorem cannot be sharpened for any of the classes W r Hk [ϕ] if the rate of approximation is characterized not by the quantities ρn (x) but rather by n−2 .

2.8 Simultaneous approximation First, let us recall some facts related with trigonometric approximation. The approximation of the derivatives of a function by the derivatives of the polynomial which approximate the function was considered by Freud [122]. He proved that, for any polynomial Tn , f (r) − Tn(r)  ≤ Cr {nr f − Tn  + En∗ (f (r) )},

(2.27)

where Cr is a constant which depends only on r. A related inequality was given by Czipszer and Freud in [78]. Theorem 2.8.1. Fix k ∈ N. (i) There exists a constant K such that, if f ∈ C k [0, 2π] and Tn ∈ Tn , then f (k) − Tn(k)  ≤ K log (1 + min{k, n}) {nr f − Tn  + En∗ (f (r) )}. Moreover, if f − Tn  ≤ CEn∗ (f ), then f (k) − Tn(k)  ≤ KCEn∗ (f (r) ).

(2.28)

2.8. Simultaneous approximation

35

(ii) There exists a constant K such that, if f, f (k) ∈ L1 [0, 2π] and Tn ∈ Tn satisfies f − Tn 1 ≤ CEn∗ (f )1 , then f (k) − Tn(k) 1 ≤ CK log (1 + min{k, n}) En∗ (f (r) )1 . (iii) For each p ∈ (1, ∞), there exists a constant Bp such that, if f, f (k) ∈ Lp [0, 2π] and Tn ∈ Tn satisfies f − Tn p ≤ CEn∗ (f )p , then f (k) − Tn(k) p ≤ CBp En∗ (f (r) )p . The inequality (2.27) was improved by Garkavi [133]. Set (k)

Cn,r (f ) = inf

max

Tn ∈Tn 1≤k≤r

f (k) − Tn  and Cn,r = sup Cn,r (f ). En∗ (f (k) ) f ∈W r (1,[0,2π])

Garkavi proved that Cn,r =

4 (ln(p + 1)) + O(ln ln ln p)), π2

where p = min{n, r} and  π Cn,r En (f (r) ). f (r) − Tn(r)  ≤ nr f − Tn  + 1 + 2 One of the first results on simultaneous approximation is due to Gelfond in 1955. Theorem 2.8.2 (Gelfond, [141]). If f ∈ C m [a, b], for n ≥ n0 , there exists Pn ∈ Pn such that  1 (k) (k) (m) 1 , (0 ≤ k ≤ m). f − Pn  ≤ C m−k ω f , n n Theorem 2.8.3 (Feinerman and Newman, [117]). There exists a constant K such that, if f ∈ C 1 [a, b], then En (f ) ≤

K En−1 (f  ) n

n ≥ 1.

(2.29)

Hasson found estimates in norms in the spirit of Garkavi’s results. Proposition 2.8.4 (Hasson, [156]). There exists a constant M with the following property: Let f ∈ C[a, b] be such that, for some λ, En (f ) ≤ λ/n, n ≥ 1, En (f ) ≤ λ. Then, if Pn is the polynomial of best approximation to f , one has Pn  ≤ M λ n,

n ≥ 1.

36

Chapter 2. The End Points Effect

Proof. Fix k such that 2k ≤ n < 2k . By differentiating the identity P = Pn − P2k +

k

(P2i − P2i−1 ) + (P1 − P0 ) + P0

i=1

and applying the Markov inequality we obtain   k

22i P2i − P2i−1  + (P1 − P0 ) P   ≤ K n2 Pn − P2k  + i=1

 2

≤K

2n E2k (f ) +

k

2

2i+1

 E2i−1 (f ) + 2E0 (f ))

i=1



 k

22(k+1) 2i+1 λ ≤K 2 λ+ 2 + 2λ) 2i−1 2k i=1   k

k i ≤ K λ 82 + 4 2 + 2) ≤ M λ n.



i=1

Theorem 2.8.5 (Hasson, [156]). Let k and r be integers. For f ∈ C r [a, b], let Pn (f ) ∈ Pn be the polynomial of best approximation for f . There exist constants M , S and T depending on r such that f (k) − Pn(k) (f ) ≤ M nk En−k (f (k) ), Pn(k) (f )

≤ f

(k)

 + M n En−k (f k

(k)

),

0 ≤ k ≤ r,

n ≥ k,

0 ≤ k ≤ r,

n ≥ k,

(2.30)

and f (k) − Pn(k) (f ) ≤ SEn−2k (f (2k) ) ≤ T En−r (f )

1 En−r (f (r) ), nr−2k

for 0 ≤ k ≤ m/2 and n ≥ m. Moreover (Roulier, [314]) f

(k)

−

Pn(k) (f )

 (r) 1 , ≤ M r−2k ω f n n 1

n > r.

Proof. It is clear that (2.30) holds for k = 0. Assume that (2.30) holds for r. By induction, for f ∈ C r+1 [0, 1] one has (k)

f (k+1) − Qn−1  ≤ M nk En−1−k (f (k+1) ),

0 ≤ k ≤ r,

n ≥ k,

where Qn−1 is the polynomial of best approximation to f  . If we set  x g(x) = f (x) − f (a) − Qn−1 (t)dt, x ∈ [a, b], a

2.8. Simultaneous approximation

37

then, for a ≤ x < y ≤ b, 

y

| g(x) − g(y) |≤

| f  (t) − Qn−1 (t) | dt ≤ En−1 (f  ) | x − y | .

x

Let Rn be the polynomials of best approximation to g. From the direct estimate and Proposition 2.8.4 we know that Rn  ≤ K1 n En−1 (f  ),

n ≥ 1,

and, using Markov inequality and (2.29), we obtain Rn(k)  ≤ Kk n n2(k−1) En−1 (f  ) ≤ Kk∗

n2k−1 En−k (f (k) (n − 1)(n − 2) · · · (n − (k − 1))

≤ Kk nk En−k (f (k) ,

0 ≤ k ≤ r + 1,

n ≥ k.

Therefore (k−1)

f (k) − Qn−1 − Rn(k)  ≤ Kk nk En−k (f (k) ) + Mr nk−1 En−k (f (k) ) ≤ Mr+1 nk En−k (f (k) ),

0 ≤ k ≤ r + 1,

n ≥ k.

x The result follows because −f (a) + a Qn−1 (t)dt + Rn (x) is the polynomial of best approximation to f . The last assertion follows from Jackson’s theorem. In fact    1 1 (r) (r) (r) 1 ≤C 1+ ω f .  En−r (f ) ≤ C ω f n−r n−r n Theorem 2.8.6 ([156]). Let a < c < d < b and let m and k be integers with 0 ≤ k ≤ m. There exists a constant C, which depends on m, c and d such that, if Pn is the polynomial of best approximation to f ∈ C m [a, b], then f (k) − Pn(k) [c,d] ≤ CEn−k (f (k) ),

n ≥ k.

Theorem 2.8.7 ([156]). Let k and r be integers, k > r ≥ 0. Fix f ∈ C r [a, b] and, for each n ∈ N, let Pn (f ) ∈ Pn be the polynomial of best approximation for f . If f is not a polynomial, there exist constants M (f, k), such that  1 Pn(k) (f ) ≤ M (f, r) n2k−r ω f (r) , n ≥ 1. n Proof. The proof of this theorem is based on an extension of f . Fix two reals c and d (c < a and b > d) and assume that f has been extended to a function F ∈ C r [c, d] in such a way that ω(F (r) , h) ≤ Cω(f (r) , h).

38

Chapter 2. The End Points Effect

Fix a sequence {Qn } of polynomials such that   K CK (k) (k) (r) 1 (r) 1 ≤ r−k ω f , , Qn − F [c,d] ≤ r−k ω F , n n n n (k)

for k ≤ r and n ≥ k + 1. One has Qn [c,d] ≤ Ck , for 0 ≤ k ≤ r and (by (k)

Bernstein’s inequality) Qn [c,d] ≤ Kk nk−r , for k > r. Since, for k ≥ 0, (k) Pn(k) [a,b] ≤ Pn(k) − Q(k) n [a,b] + Qn [a,b]

and

 Kl 2k 2k (r) 1 E  ≤ S n [P − Q  ≤ N n (f ) + ω(f , , Pn(k) − Q(k) k n n [a,b] k n [a,b] n nr n

Jackson’s inequality yields Pn(k) [a,b]

≤ C5 n

2k−r

   (r) 1 ω f , + max Kk , Kk nk−r . n

Thus the proof finishes by proving that the second term can be estimated with the first one.  Trigub was one of the first in considering a point-wise estimate for simultaneous approximation by algebraic polynomials. He also noticed that we can use the second-order modulus, instead of the first one, and provided some inequalities for the derivatives of the polynomials. In 1968 Malozemov [246] proved that the constant in the corresponding estimates of Gelfond and Trigub do not depend on the functions. We present the assertion as it appeared in a paper of Malosemov [245]. Theorem 2.8.8 (Trigub, [388]). If f ∈ C r [−1, 1], then for each n ∈ N there exists a polynomial Pn ∈ Pn such that, for all x ∈ [−1, 1] and k = 0, 1, . . . , r,   r−k w f (r) , Δn (x) (2.31) | f (k) (x) − Pn(k) (x) |≤ Cr (Δn (x)) where Cr does not depend upon n or f . Is the last result a consequence of the particular polynomials used in the approximation? In 1966 Teliakovskii showed that, for a differentiable function f , the derivatives of any sequence of the polynomials which approximate f with the rate given in Timan’s theorem, approximate f  with a similar rate. We need an estimate for the derivatives of polynomials. Proposition 2.8.9. There exists a constant R with the following property: let a ≥ 0 be a real number, r ≥ 1 an integer and ω a modulus of continuity. If a polynomial Pn satisfies the inequality | Pn (x) |≤ (Δn (x))r ω(Δn (x)) + a, then

x ∈ [−1, 1],

  | Pn (x) |≤ R (Δn (x))r−1 ω(Δn (x)) + a(Δn (x))−1 ,

x ∈ [−1, 1].

2.8. Simultaneous approximation

39

The last result was proved by Lebed [225] in the case a = 0. Other proofs were given in [107] and [379] (p. 219–226). According to Teliakovskii [371], for a > 0 the proof can be obtained with arguments similar to the one used in [379]. Theorem 2.8.10 (Teliakovskii, [371]). Assume r ∈ N0 and f ∈ C r [−1, 1]. If {Pn (f, x)} is a sequence of polynomials satisfying (2.9), then for k = 1, . . . , r,   r−k ω f (r) , Δn (x) , | f (k) (x) − Pn(k) (f, x) |≤ Cr,k (Δn (x)) where the constant Cr,k does not depend upon f or n. Proof. We only present the main ideas of the proof. Let {Pn } be a sequence of polynomials for which the Timan estimate (2.9) holds. For s ∈ N0 , write ns = 2s n, p0 = Pn and ps = Pns . From the identity f (x) − p0 (x) =

∞

[ps (x) − ps−1 (x)]

s=1

we obtain

∞  

   (k) |=  [p(k) | f (k) (x) − s (x) − ps−1 (x)]   s=1     √ r−k √ ∞ 2

1 − x 1 − x2 1 1 ≤ (RA + R) + ω f (r) , + n ns n ns s s s=1     √ √ r−k ∞ 2

1 − x 1 1 1 − x2 + s 2 + s 2 . ≤ (RA + R) ω f (r) , sn 2 4 n 2s n 4 n s=1 (k) p0 (x)

If k < r, then | f (k) (x) − Pn(k) (x) |≤ C (Δn (x))

r−k

∞  

ω f (r) , Δn (x) s=1

1 2s(r−k)

.

The theorem is proved for k < r. For the case k = r, it is sufficient to consider the case r = 1. Assume r = 1 and fix a point x0 and set h = Δn (x0 ). There exists a function Fh (f ) ∈ C 1 [−1, 1] such that 1 h ω(f  , h), 2 | f  (x) − Fh (f, x) | ≤ ω(f  , h) | f (x) − Fh (f, x) | ≤



and ω(Fh , t)

≤

δ ω(f  , h)/h,

if

δ ≤ h,

3ω(f  , h),

if

h < δ.

(2.32) (2.33)

(2.34)

40

Chapter 2. The End Points Effect

Then

| f  (x) − pn (x) | ≤| f  (x) − Fh (x) | + | Fh (x) − pn (x) | ≤ ω(f  , h)+ | Fh (x) − pn (x) | .

(2.35)

There exist polynomials Qm such that | Fh (x) − Qm (x) |≤ C Δm (x) ω(Fh , Δm (x)).

(2.36)

Now, we use the representation Fh (x) − pn (x) =

∞

[Qns (x) − Qns−1 (x)] + Qn (x) − pn (x).

(2.37)

s=1

In this case we have | Qns (x) − Qns−1 (x) |≤ C Δns (x) ω (Fh , Δns (x)) . From Proposition 2.8.9 (with a = 0) we obtain | Qns (x) − Qns−1 (x) |≤ C1 ω (Fh , Δns (x)) . On the other hand, we can use (2.36), (2.32), the hypothesis (2.9) and (2.34) to estimate the difference Qn − pn . In fact | Qn (x) − pn (x) | ≤| Qn (x) − Fh (x) | + | Fh (x) − f (x) | + | f (x) − pn (x) | 1 ≤ C Δn (x) ω(Fh , Δn (x)) + hω(f  , h) + AΔn (x)ω(f  , Δn (x)) 2 1  ≤ C2 Δn (x) ω(f , Δn (x)) + hω(f  , h). 2 From the last estimate and Proposition 2.8.9 (with a = hω(f  , h)/2) we obtain | Qn (x) − pn (x) |≤ C3 ω(f  , Δn (x)) + C4 (Δn (x))−1 hω(f  , h). Therefore the series in (2.37) converges uniformly and we can differentiate term by term. That is ∞ 

    −1  | Fh (x)−pn (x) |≤ C1 ω(Fh , Δns (x)) + ω(f , Δn (x)) + (Δn (x)) hω(f , h) . s=1

Finally, for x = x0 and h = Δn (x0 ), from the last inequality and (2.35) one has

 ∞

Δns (x0 )ω(f  , h) hω(f  , h)  + ω(f , Δn (x0 )) + |≤C h Δn (x0 ) s=1    ∞ 1 1 − x20 ω(f  , h)

 + s 2 ≤ C ω(f , Δn (x0 )) + h 2s n 4 n s=1 



| f (x0 ) −

pn (x0 )

≤ Cω(f  , Δn (x0 )).



2.8. Simultaneous approximation

41

In particular, from Timan’s theorem Teliakovskii derived a new proof of the Trigub result presented above. On the other hand, Theorem 2.8.10 can be obtained from Theorem 2.8.8 and Proposition 2.8.9. An analogue of Theorem 2.8.10, with δn (x) instead of Δn (x) is due to Gopengauz. He constructed linear polynomial operators Ln,r : C r [−1, 1] → Pn for each fixed r ≥ 0, such that the following theorem holds: Theorem 2.8.11 (Gopengauz, [150]). For each r ≥ 0 there exists a sequence of linear operators Ln,r : C r [−1, 1] → Pn (n ≥ 4r + 5) such that, for f ∈ C r [−1, 1] for 0 ≤ k ≤ r,   r−k (r) (x) |≤ C (δ (x)) ω f , δ (x) , (2.38) | f (k) (x) − L(k) r n n n,r where the constant Cr does not depend on f , n and x. In 1978 V´ertesi noticed that, under additional assumptions, one can replace Δn (x) by δn (x). Theorem 2.8.12 (V´ertesi, [399]). Assume r ∈ N0 and f ∈ C r [−1, 1]. If {Pn (f, x)} is a sequence of polynomials satisfying (2.9) and Pn(k) (f, ±1) = f (k) (±1),

(k = 0, 1, . . . , r),

then for k = 0, 1, . . . , r, | f (k) (x) − Pn(k) (f, x) |≤ Cr,k (δn (x))

r−k

  ω f (r) , δn (x) ,

where the constant Cr,k does not depend on f or n. There are other similar inequalities due to Gonska and Hinnemann. Theorem 2.8.13 (Gonska and Hinnemann, [147]). Fix an integer r ≥ 0, a constant Cr and let Ln : C[−1,1] → Pn (n ≥ r) be a sequence of linear operators such that, for every x ∈ [−1, 1] and f ∈ C r [−1, 1], (i) (ii)

Ln (f ) ≤ Cr f , f ∈ C[−1, 1], | f (x) − Ln (f, x) | ≤ Cr (Δn (x))r f (r) .

Then, there exists a constant Dr such that, for each 0 ≤ k ≤ r and f ∈ C r [−1, 1], (k) . L(k) n (f ) ≤ Cr f

Theorem 2.8.14 ([147]). Fix r ≥ 0, s ≥ 1 and let Cr and Cr,s be constants. (i) There exists a constant Dr such that, if f ∈ C r [−1, 1] and Pn ∈ Pn (n ≥ r) satisfies | f (x) − Pn (x) | ≤ Cr (Δn (x))r f (r) , then for 0 ≤ k ≤ r | f (k) (x) − Pn(k) (x) | ≤ Dr (Δn (x))r−k f (r).

42

Chapter 2. The End Points Effect

(ii) There exists a constant Mr,s such that, if f ∈ C r [−1, 1] and for Pn ∈ Pn (n ≥ r + s) one has | f (x) − Pn (x) | ≤ Cr,s (Δn (x))r ωs (f (r) , Δn (x)), then, for 0 ≤ k ≤ r, | f (k) (x) − Pn(k) (x) | ≤ Mr,s (Δn (x))r−k ωs (f (r) , Δn (x)). The Hasson results (Theorem 2.8.5 and 2.8.6) involve estimates in norm. In [234] Leviatan found point-wise estimates in the spirit of the results of Timan and Trigub, but considering the best approximation instead of the modulus of smoothness of the derivatives. Theorem 2.8.15 (Leviatan, [234]). For r ≥ 0 let f ∈ C r [−1, 1] and let Pn ∈ Pn denote its polynomial of best approximation on [−1, 1]. Then for each 0 ≤ k ≤ r and every −1 ≤ x ≤ 1, | f (k) (x) − Pn(k) (x) | ≤

Cr [Δn (x)]−k En−k (f (k) ), nr

n ≥ k,

| f (k) (x) − Pn(k) (x) | ≤

Cr [Δn (x)]−k En−r (f (r) ), nr

n ≥ k,

and

where Cr is an absolute constant which depends only on r. Proof. For k = 0 the result is evident. Assume that it is true for r. By induction, for f ∈ C r+1 [0, 1] one has (k)

| f (k+1) (x) − Qn−1 (x) | ≤

M (Δn (x))−k En−1−k (f (k+1) ), nk

0 ≤ k ≤ r,

where Qn−1 is the polynomial of best approximation to f  . If we set  g(x) = f (x) −

x

−1

Qn−1 (t)dt = f (x) − Qn (x),

x ∈ [−1, 1],

then, | g  (x)) |≤ CEn−1 (f  ). There exists a polynomial Sn such that g − Sn  ≤

C En−1 (f  ), n

and

Sn  ≤ C En−1 (f  ).

Thus, from (iii) of Theorem 2.7.4 one has | Sn(k) (x) | ≤ C(Δn (x))1−k Sn  ≤ C1 (Δn (x))1−k En−1 (f  ).

n ≥ k,

2.8. Simultaneous approximation

43

Let Rn be the polynomial of best approximation to g. Using again (iii) of Theorem 2.7.4 and taking into account that En (g) = En (f ), one has | Rn(k) (x) − Sn(k) (x) | ≤ C(Δn (x))−k Rn − Sn  ≤ C(Δn (x))−k (En (g) + g − Sn ) ≤ C1

(Δn (x))−k En−1 (f  ). n

Therefore | Rn(k) (x) | ≤ C2

(Δn (x))−k (Δn (x))−k En−1 (f  ) ≤ C3 En−k (f (k) ). n nk

Since Pn = Qn + Rn is the polynomial of the best approximation of f we have the result.  For the last theorem some of the results of Hasson are easily derived. Theorem 2.8.16 ([234]). For r ≥ 0 let f ∈ C r [−1, 1] and let n ≥ r. Then there exists a polynomial Pn ∈ Pn such that | f (k) (x) − Pn(k) (x) | ≤ Cr [Δn (x)]r−k En−r (f (r) ),

n ≥ k,

(2.39)

for k = 0, 1, . . . , r and −1 ≤ x ≤ 1. Kilgore combined the estimates of Gopengauz and Leviatan. Theorem 2.8.17 (Kilgore, [191]). If f ∈ C m [−1, 1], for each n > 2m, there exists a polynomial Pn ∈ Pn such that, for k = 0, 1, . . . , m, m−k √ 1 − x2 (k) (k) En−m (f (m) ), (2.40) | f (x) − Pn (x) |≤ C(m, k) n where the constants C(m, k) depend only on m and k. An algebraic analog of the result of Czipszer and Freud in [78] is the following. Theorem 2.8.18 (Kilgore and Szabados, [196]). Let g ∈ C q [−1, 1] be such that g (k) (±1) = 0 for k ≤ q − 1. Let ε > 0 and assume there is a sequence {Pn+q } (Pn+q ∈ Pn+q ) such that    g(x) − Pn+q (x)  ε   √  ( 1 − x2 )q  ≤ nq . Then, for |x| ≤ 1 and k ≤ q, q−k    √ 2    (k)  1 − x 1 (q)   g(x) − pn (x) ≤  δk,q inf (g − pn ) + γk,q ε , + 2   pn n n where δk,q and γk,q ε depend on k and q.

44

Chapter 2. The End Points Effect

2.9 Zamansky-type estimates As we see in Theorem 1.2.1 concerning trigonometric approximation, for σ < s, the (s) conditions En∗ (f ) = f − Tn  = O(n−σ ) and Tn  = O(n−(σ−s) ) are equivalent. As Hasson showed there is not a direct analogue in the algebraic case. Theorem 2.9.1 (Hasson, [156]). There exists a function f ∈ C[−1, 1] such that En (f ) ≤ K/n and, if Pn is the polynomial of best approximation to f on [−1, 1], Pn [a,b] > K log n, n ∈ N, whenever −1 < a < b < 1. (k)

Leviatan also studied the growth of the sequence {Pn }. His proof is based in a theorem of Runck. Theorem 2.9.2 (Runck, [315]). For r ≥ 0 let f ∈ C r [−1, 1] and let n ≥ r. Then there exists a polynomial Pn ∈ Pn such that | f (k) (x) − Pn(k) (x) | ≤ Ck [Δn (x)]r−k ω(f (r) , Δn (x)), and

  | Pn(k) (x) | ≤ Cr [Δn (x)]r−k ω f (r) , Δn (x) ,

0≤k≤r k ≥ r + 1,

with constant independent of f . Theorem 2.9.3 (Leviatan, [234]). For r ≥ 0 let f ∈ C r [−1, 1] and let Pn ∈ Pn , denote its polynomial of best approximation on [−1, 1]. Then for each k > r there exists a constant K, depending only on k, such that, for every −1 ≤ x ≤ 1,  K (k) −k (r) 1 | Pn (x) | ≤ r [Δn (x)] ω f , , n ∈ N. n n This improves some results of Hasson. In particular, for k > r.  1 , Pn(k)  ≤ K n2k−r ω f (r) , n where the constant K depends only on k. An extension to an estimate with higherorder moduli is given as follows. Theorem 2.9.4 ([234]). For r ≥ 1, let f ∈ C[−1, 1] and let Pn ∈ Pn , denote its polynomial of best approximation on [−1, 1]. Then for each k ≥ r there exists a constant K depending on k and r, such that for every −1 ≤ x ≤ 1,  1 n ∈ N. (2.41) | Pn(k) (x) | ≤ K [Δn (x)]−k ωr f, n There is also a nice remark of Leviatan in the paper quoted above: the upper bound of the K-functional in the characterization of the usual modulus of continuity can be given by polynomials. That is, for every f ∈ C[−1, 1] and n ∈ N

2.9. Zamansky-type estimates

45

there is a polynomial Pn ∈ Pn such that  1 and f − Pn  ≤ C ωr f, n

Pn(r) 

 1 . ≤ C n ωr f, n r

In 1985 Ditzian [95] improved (2.41) by proving a similar inequality but in terms of so-called Ditzian-Totik moduli (the definition will be given below). That is  1 (k) −k ϕ | Pn (x) | ≤ K [Δn (x)] ωr f, . n (k)

The results are the best possible. If | Pn (x) | ≤ K [Δn (x)]−k ψ(n) where ψ(n) is decreasing, ψ(n) = o(1), and satisfies some additional conditions, then ωrϕ (f, 1/n) ≤ M ψ(n). This provides the analogue to the Sunouchi-Zamanski theorem. Theorem 2.9.5 (Ditzian, [95]). If for some integer r and decreasing sequence ψ(n), l

2kr ψ(2k ) ≤ M 2lr ψ(2l )

and

En (f ) ≤ ψ(n),

k=1

then for Pn , the polynomial satisfying f − Pn  = En (f ), one has | Pn(k) (x) | ≤ K [Δn (x)]−k ψ(n). In particular, if for some r, l

2kr E2k (f ) ≤ M 2lr E2l (f )

k=1

then

| Pn(k) (x) | ≤ K [Δn (x)]−k En (f ). Another extension is due to Shevchuk.

Theorem 2.9.6 (Shevchuk, [338]). If f ∈ C r [−1, 1] and ωk (f (r) , t) ≤ ω(t) (0 < t ≤ 1/k), then for any n ≥ r + k − 1 there exists Pn ∈ Pn such that, for all x ∈ [−1, 1], | f (j) (x) − Pn(j) (x) | ≤ C(Δn (x))r−j ω(Δn (x)),

0 ≤ j ≤ r,

and | Pn(j) (x) | ≤ C(Δn (x))r−j ω(Δn (x)) + C(r + k − j)(Δn (x))−j f x,n, for 0 ≤ j ≤ r + k, where f x,n = max {| f (u) | : u ∈ [x − Δn (x), x + Δn (x)] ∩ [−1, 1]}.

46

Chapter 2. The End Points Effect

2.10 Fuksman-Potapov solution to the second problem In the last sections we have seen theorems which provide characterization for certain classes of functions. For instance, Theorem 2.7.3 characterizes functions satisfying ω(f (r) , t) ≤ Cω(t) by means of its approximations for algebraic polynomials. In Theorems 2.7.5 and 2.7.7 similar results were presented for functions satisfying f (r) ∈ Lipα [−1, 1] or in the Zygmund class respectively. These theorems provide the analogue of the first interpretation after (1.11). That is, we have a characterization of functions satisfying a classical Lipschitz condition in terms of the rate of pointwise approximation by algebraic polynomials. Let us consider the problem of characterization of other classes of functions. For r ∈ N and α ∈ (0, 1), let K(r, α) = {f ∈ C[−1, 1] : En (f ) ≤ M (f )n−r−α }. Classes K(r, α) are defined in terms of the rate of convergence of the best approximation. The classes C r,α [−1, 1] and K(r, α) are different. For instance, for √ f (x) = 1 − x2 one has, f ∈ K(0, 1) but, for any δ > 1/2, f ∈ / C 0,δ [−1, 1]. It was an interesting question to describe classes K(r, α) without any reference to approximation by polynomials. One of the first results in this direction is due to Fuksman [129]. For f ∈ C r (−1, 1) and 0 ≤ k ≤ r/2, let ψk (x) = f (r−k) (x)(1 − x2 )r/2−k and consider the condition  sup | ψk (x) − ψk (x + h) | ≤ C h ∈ Λ(x, δ)

δ √ √ 1 − x2 + δ

α ,

(2.42)

where Λ(x, δ) = {h :| h |≤ δ, | x + h |≤ 1}. We assume ψ1 (1) = ψ1 (−1) = 0 for odd k. Let S(r, α) = {f ∈ C r (−1, 1) : ψk ∈ C[−1, 1] (0 ≤ k ≤ r/2) and (2.42) holds }. Theorem 2.10.1 (Fuksman, [129]). For each r ∈ N0 and 0 < α < 1, one has K(r, α) = S(r, α). Proof. In order to verify the inclusion S(r, α) ⊂ K(r, α), for f ∈ S(r, α), define F (t) = f (cos(t)). If r = 0, then ψ0 (x) = f (x) and (2.42) yields  | f (x + h) − f (x) | ≤ C

δ √ √ 1 − x2 + δ



α ≤ C min

δ √ 1 − x2

α    √ α , δ

for h ∈ λ(x, δ). Set h = cos(t + h) − cos t and δ =| h sin t | +h2 . Since | cos(t + h) − cos t |=| cos t(1 − cos h) + sin t sin h |≤ δ,

2.10. Fuksman-Potapov solution to the second problem

47

one has  | F (t + h) − F (t) | ≤ C min

δ √ 1 − cos2 t

α    √ α . , δ

(2.43)

We should consider two cases. Case 1. If | h |≤| sin t |, then δ | h sin t | +h2 = ≤ 2|h|. | sin t | | sin t | Case 2. If | h |>| sin t |, then √  √ δ = | h sin t | +h2 ≤ | h | 2 ≤ 2 | h | . Therefore | F (t + h) − F (t) | ≤ C1 | h |α . Now we consider that r > 0. By induction with respect to r it can be proved that there exists trigonometric polynomials ϕi,r ∈ Tr−i such that [r/2]

F (r) (t) =



r

f (r−i) (cos t) sinr−2i (t) ϕi,k (t) +

i=0

f (r−i) (cos t) ϕi,k (t).

i=[r/2]+1

But f (r−i) (cos t) | sinr−2i (t) |= ψi (cos t), then we can write [r/2]

F (r) (t) =

i=0

r

Ψi (t) ϕi,k (t) +

f (r−i) (cos t) ϕi,k (t),

i=[r/2]+1

where Ψi (t) = f (r−i) (cos t)sign(sin t)k . It can be proved that these functions are continuous. Moreover, as in the proof of the case r = 0, each function Ψi satisfies a Lipschitz condition of order α. Therefore, there exist a constant C and a sequence {Tn } of even trigonometric polynomials such that | F (t) − Tn (t) |≤ Cn−(k+α) . By taking Pn (x) = Tn (arccos x) we conclude that f ∈ K(r, α). Let us consider the relation K(r, α) ⊂ S(r, α). Fix f ∈ K(r, α) and a sequence {Pn } of polynomials such that f − Pn  ≤ Cn−(k+α) . If we set Qn = P2n − P2n−1

(n ≥ 1),

(2.44)

then Qn  ≤ C2−n(r+α)

(2.45)

48

Chapter 2. The End Points Effect

and take into account that f (x) =

∞

Qn (x),

n=1

then f (i) (x) =

∞

Q(i) n (x),

(i = 0, 1, . . . , r).

n=1

Set ψj (x) = f (r−j) (x)(1 − x2 )r/2−j , then ∞

ψj (x) =

Q(r−j) (x)(1 n

2 r/2−j

−x )

=

n=0

m

+

n=0

∞

= Lm (x) + L∗m (x),

(2.46)

n=m+1

where m will chosen later. We will estimate the modulus of continuity of Lm and L∗m . First m  

 d  2 r/2−j r−j  Qn (u)  | Lm (x + h) − Lm (x) |≤| h |  du (1 − u ) u=x+hθ n=0 m    

    2 r/2−j r−j−1 ≤| h | (u) + − u ) Q (u) 2u(1 − u2 )r/2−j−1 Qr−j  (1  n n n=0

. u=x+hθ

Now we have two different estimates: taking into account (2.45) and (2.23) (with l = r − j, q = r − 2j − 1 p = j + 1 (p + q = l)) one has m

  2u(1 − u2 )r/2−j−1 Qr−j n (u)

u=x+hθ

n=0

≤ C1

m

2

(r+1)n

2 r/2−j−1

Qn (1 − u )

(1 − u )

n=0

≤ C2 (1 − (x + hθ)2 ))−1/2

m

   

2 −r/2+j+1/2 

2(r+1)n 2−n(r+α) ≤ C3

n=0

u=x+hθ

2(1−α)m . (1 − (x + hθ)2 ))1/2

On the other hand, (2.23) (with l = r − j, q = r − 2j − 2 p = j + 2 (p + q = l)) one has m 

 2u(1 − u2 )r/2−j−1 Qr−j (u)  n u=x+hθ

n=0

≤ C1

m

n=0

≤ C2

m

n=0

   2(r+2)n Qn (1 − u2 )r/2−j−1 (1 − u2 )−r/2+j+1   2(r+2)n 2−n(r+α) ≤ C3 2(2−α)m .

u=x+hθ

2.10. Fuksman-Potapov solution to the second problem

49

Since for the other term in the estimate of | Lm (x + h) − Lm (x) | we can obtain similar inequalities, we have proved that | Lm (x + h) − Lm (x) | ≤ C4

2(1−α)m (1 − (x + hθ)2 ))1/2

(2.47)

and | Lm (x + h) − Lm (x) | ≤ C5 2(2−α)m .

(2.48)

With similar arguments we also prove that | L∗m (x + h) − L∗m (x) |≤ C6 2−αm .

(2.49) 

If ε > 0, | h |≤ ε and | x |≤ 1−ε, we take m such that 2m < (1 − ε)2 − x2 = r ≤ 2m+1 . Then from (2.46), (2.47) and (2.49) we obtain   α   α  1−α 1 r r |h| + = 2C . | ψj (x + h) − ψj (x) |≤ C | h | r |h| |h| r If we take m such that 2m < (| h |)−1/2 ≤ 2m+1 , then from (2.46), (2.48) and (2.49) we obtain ⎛ α ⎞ 2−α   1 1 ⎠ = 2C (| h |)α/2 . +  | ψj (x + h) − ψj (x) |≤ C ⎝| h |  |h| |h| Thus, if h ∈ Λ(x, ε), then

  | ψj (x + h) − ψj (x) | ≤ C min (| h | /r)α , | h |α/2 α  = C | h |α min (1/r), | h |−1/2  α C1 | h |α |h|   ≤ = C1 (r + | h |)α r+ |h|  α  α ε ε √ ≤ C1 ≤ C2 √ , √ r+ ε 1 − x2 + ε

since

1 6 1 √ =  √ . √ ≤ √ 2 2 r+ ε 1 − x2 + ε (1 − ε) − x + ε

Finally, if r is odd, from (2.49) we know that the series (2.46) converges uniformly on [−1, 1]. Moreover, for j < [r/2], one has k > 2j, thus ψj (±1) = 0. We have proved that f ∈ S(r, α).  The result also can be extended to the case when  M (f ) 1 ω En (f ) ≤ nk n

50

Chapter 2. The End Points Effect

where ω is a modulus of continuity satisfying conditions (2.22). In this case, the definition of the class S(r, ω) is similar to the one of S(r, k), but condition (2.42) is replaced by  δ √ sup | ψk (x) − ψk (x + h) |≤ C ω √ . 1 − x2 + δ (h, x) ∈ Λ(δ) Theorem 2.10.2 (Fuksman, [129]). Let f ∈ C[−1, 1], r a positive integer and α ∈ (0, 1). The following assertions are equivalent: (i) f ∈ C 2r (−1, 1) and, for 0 ≤ k ≤ r and ψk (x) = f (r−k) (x)(1 − x2 )r−k , one has ψk ∈ C[−1, 1] and (2.42) holds. (ii) For each n ∈ N there exists an algebraic polynomial Pn ∈ Πn−1 such that, for each x ∈ [−1, 1], D | f (x) − Pn (x) | ≤ 2r+α n where D is a positive constant which does not depend on x or n. In 1980 Potapov [295] unified the results of Dzyadyk and Fuksman. He proved an analogue to Theorem 2.10.3, but with the condition α + β < 1 instead of α + β/2 < 1. Notice that, by taking β = 0 we obtain the Dzyadyk characterization and for β = −α the Fuksman result. The results we present here were proved by Potapov in 2005 [303]. Theorem 2.10.3 (Potapov, [303]). Fix reals α and β such that α ∈ (0, 1), α + β ≥ 0 and α + β/2 < 1. For f ∈ C[−1, 1] the following assertions are equivalent: (i) For each x ∈ [−1, 1], one has sup { h : |h|≤δ, |x+h|≤1}

| f (x + h) − f (x) |≤ C1 δ α



1 − x2 +

√ β δ ,

where C1 is a positive constant which does not depend on δ or x; (ii) For each n ∈ N there exists Pn−1 ∈ Πn−1 such that, for each x ∈ [−1, 1], | f (x) − Pn−1 (x) | ≤ C2 nβ (Δn (x))

α+β

where C2 is a positive constant which does not depend on x or n. Proof. (i) =⇒ (ii). Fix m, s ∈ N such that (n − 1)/s < m ≤ 1 + (n − 1)/s and define  π f (cos(t + y))Km,s (t)dt, Q(x) = −π

where x = cos y and where K2q is given by (2.8) with s = q. There exist positive constant C1 and C2 such that  π | t |β Km,s (t)dt ≤ C2 m−β . C1 m2s−1 ≤ cm,s ≤ C2 m2s−1 and C1 mβ ≤ −π

Moreover, Qm ∈ Pn−1 .

2.10. Fuksman-Potapov solution to the second problem

51

Since for | t |≤ π, one has   γ =| t | 1 − x2 + t2 )α ≤ 3(| t | 1 − x2 + t2 ), as in the proof of (2.43) from (i) we have 

1 − x2 + t2 )α (γ)  ≤ 3C | t |α ( 1 − x2 + t2 )α+β  ≤ C1 (| t |α ( 1 − x2 )α+β + | t |2α+β ).

| f (cos(y + t)) − f (cos y) | ≤ C (| t |

Now one has



β

π

| f (x) − Q(x) |≤ | f (cos(t + y)) − f (cos y) | Km,s (t)dt −π   π  π α+β α 2α+β 2 | t | Km,s (t)dt + |t| Km,s (t)dt ≤C ( 1−x ) −π −π  √  ( 1 − x2 )α+β 1 + 2α+β ≤ C2 nβ (Δn (x))α+β . ≤ C1 mα m We have proved (ii). (ii) =⇒ (i). We should modify the arguments of the proof of Theorem 2.10.1. If Qn be defined by (2.44), then | Qk (x) |≤ C2k β (Δ2k (x))α+β and from Theorem 2.7.5 we obtain | Qk (x + h) − Qk (x) |≤ C1 | h | 2kβ ( Δ2k (x + hθ) )α+β−1 . Fix x ∈ [−1, 1] and | x + h |≤ 1. Fix N ∈ N which will be chosen later. Notice that | Δh f (x) | ≤| f (x) − P2N (x) | + | f (x + h) − P2N (x + h) | +

| Qk (x + h) − Qk (x) |

k=0

 ≤ C3

N

2

Nβ

α+β

((Δ2N (x))

α+β

+ (Δ2N (x + h))

)+ | h |

N

( Δ2k (x + hθ) )α+β−1 k=0

2−kβ

  ≤ C3 2N β (Δ2N (x))α+β + (Δ2N (x + h))α+β )+ | h | (Δ2N (x + hθ))α+β−1 , where the sum is estimated as follows. If α + β ≤ 1 and α < 1, then 2k(α+β−1) (Δ2k (x + hθ))α+β−1 ≤ 2N (α+β−1) (Δ2N (x + hθ))α+β−1 .



52

Chapter 2. The End Points Effect

Hence N

( Δ2k (x + hθ) )α+β−1 k=0

2−kβ

≤ 2N (α+β−1) (Δ2N (x + hθ))α+β−1

N

2k(1−α)

k=0

≤ C 2N β (Δ2N (x + hθ))α+β−1 . On the other hand, if α + β > 1 and α + β/2 < 1, then  N α+β−1 2 k(α+β−1) α+β−1 (Δ2k (x + hθ)) ≤ 2N (α+β−1) (Δ2N (x + hθ))α+β−1 . 2 2k Hence N

( Δ2k (x + hθ) )α+β−1 k=0

2−kβ

≤2

2N (α+β−1)

α+β−1

(Δ2N (x + hθ))

N

2k(2−2α−β)

k=0

≤ C 22N (α+β−1) (Δ2N (x + hθ))α+β−1 2N (2−2α−β) = C 2N β (Δ2N (x + hθ))α+β−1 . To finish the proof we should choose N . Case 1. Suppose that 0 < h < 1/4 and x ∈ [−1, −1 + 2h] ∪ [1 − 2h, 1 − h]. Chose N such that 2−2N −1 ≤ h < 2−2N . Then 1 − x2 ≤ 1 − (1 − 2h)2 ≤ 4h ≤ 4 2−2N and

1 − (x + hθ)2 ≤ 1 − x2 + 2h ≤ 6h ≤ 6 2−2N .

Hence 2−N ≤ 2−N ≤ 2−N ≤ and



1 − x2 + 2−N ≤ 32−N ,



1 − (x + h)2 + 2−N ≤ 42−N ,



1 − (x + hθ)2 + 2−N ≤ 4 2−N ,

 √   √ 1 √ h + 1 − x2 ≤ h ≤ h + 1 − x2 , 3

and we obtain C (2−N (α+β) + h2N 2−N (α+β−1) ) 2N α  √ ≤ C1 2−N (2α+β) ≤ C2 hα+β/2 ≤ C3 hα ( 1 − x2 + h)β .

| f (x + h) − f (x) | ≤

Case 2. Suppose that 0 < h < 1/4 and x ∈ [−1 + 2h, 1 − 2h]. Choose N such that √ √ 1 − x2 1 − x2 1 − x2 , then   1 1 − x2 ≤ 1 − (x + hθ)2 + N . 2 Hence, in this case    1 1 1 1 − x2 ≤ 1 − (x + hθ)2 + N ≤ 2 1 − x2 + N ≤ 4 1 − x2 . 2 2 2 With these inequalities we obtain   C  | f (x + h) − f (x) | ≤ N α 1 − x2 )α+β + h2N ( 1 − x2 )α+β−1) 2  √ C2  ≤ N α ( 1 − x2 )α+β ≤ C3 hα ( 1 − x2 + h)β . 2 Case 3. The case h ∈ (−1/4, 0) can be treated as the case h ∈ (0, 1/4). Case 4. For δ < 1/4 the proof follows from the arguments given above. For δ ≥ 1/4 the proof is simple. 

54

Chapter 2. The End Points Effect

The Chebyshev differential operator is defined by D(f, x) = (1 − x2 )f  (x) − xf  (x). Moreover, set D(1) = D and D(r) = D(D(r−1) ), for r ≥ 2. Theorem 2.10.4 (Potapov, [303]). Fix real numbers σ ≥ 0 and γ > 0. For f ∈ C[−1, 1], the following assertions are equivalent: (i) For each n ∈ N (n ≥ 2) there exists Pn−1 ∈ Πn−1 such that, for each x ∈ [−1, 1], C1 | f (x) − Pn−1 (x) | ≤ 2+γ−σ (Δn (x))σ n where C1 is a positive constant which does not depend on x or n. (ii) For any interval [a, b] ⊂ (−1, 1), f ∈ C 2 [a, b], Df ∈ C[−1, 1] and for each n ∈ N (n ≥ 2) there exists Rn−1 ∈ Πn−1 such that, for each x ∈ [−1, 1], | Df (x) − Rn−1 (x) | ≤

C2 σ (Δn (x)) nγ−σ

where C3 is a positive constant which does not depend on x or n. (iii) For any interval [a, b] ⊂ (−1, 1), f ∈ C 2 [a, b], f  (x), (1 − x2 )f  (x) ∈ C[−1, 1] and for each n ∈ N (n ≥ 2) there exists Qn−1,1 , Qn−1,2 ∈ Πn−1 such that, for each x ∈ [−1, 1], | f  (x) − Qn−1,1 (x) | ≤ C3 nσ−γ (Δn (x))

σ

| (1 − x2 )f  (x) − Qn−1,2 (x) | ≤ C1 nσ−γ (Δn (x))

σ

and

where C3 is a positive constant which does not depend on x or n. Theorem 2.10.5 ([303]). Fix real numbers σ ≥ 0 and γ > 0. For f ∈ C[−1, 1], the following assertions are equivalent: (i) For each n ∈ N (n ≥ 2) there exists Pn−1 ∈ Πn−1 such that, for each x ∈ [−1, 1], C1 | f (x) − Pn−1 (x) | ≤ γ−σ (Δn (x))σ+1 n where C3 is a positive constant which does not depend on x or n. (ii) f ∈ C 1 [−1, 1] and for each n ∈ N (n ≥ 2) there exists Rn−1 ∈ Πn−1 such that, for each x ∈ [−1, 1], | f  (x) − Rn−1 (x) | ≤

C1 σ (Δn (x)) nγ−σ

where C3 is a positive constant which does not depend on x or n.

2.10. Fuksman-Potapov solution to the second problem

55

Theorem 2.10.6 ([303]). Let f ∈ C[−1, 1], r and ρ be non-negative integers and fix α and β such that α ∈ (0, 1), α + β ≥ 0 and α + β/2 < 1. The following assertions are equivalent: (i) f ∈ C 2ρ+r (−1, 1), ψ(x) = D(ρ) f (r) (x) ∈ C[−1, 1] and sup { h : |h|≤δ, |x+h|≤1}

| ψ(x) − ψ(x + h) |≤ C1 δ α



1 − x2 +

√ β δ ,

where C1 is a positive constant which does not depend on δ or x; (ii) f ∈ C 2ρ+r (−1, 1) and, for 0 ≤ k ≤ ρ and ψk (x) = f (2ρ+r−k) (x)(1 − x2 )ρ−k , one has ψk ∈ C[−1, 1] and sup { h : |h|≤δ, |x+h|≤1}

| ψk (x) − ψk (x + h) |≤ C2 δ α



1 − x2 +

√ β δ ,

where C2 is a positive constant which does not depend on δ or x; (iii) For each n ∈ N there exists an algebraic polynomial Pn−1 ∈ Πn−1 such that, for each x ∈ [−1, 1], | f (x) − Pn−1 (x) | ≤

C3 2ρ−β n

(Δn (x))r+α+β

where C3 is a positive constant which does not depend on x or n. Proof. Assume condition (iii) holds. Then there exists a sequence {Pn } of algebraic polynomials for which | f (x) − Pn−1 (x) |≤

C n2ρ−β

(Δn (x))r+α+β .

By applying ρ-times Theorem 2.10.5 we obtain that condition (iii) is equivalent to the following condition A: there exists a sequence {Rn } (n ≥ 2) of algebraic polynomials Rn ∈ Πn such that, for each x ∈ [−1, 1], | f (r) (x) − Rn (x) | ≤

C (Δn (x))α+β . n2ρ−β

By applying ρ-times Theorem 2.10.4 (which is equivalent to condition (i) and (ii)) we obtain that condition A is equivalent to the following condition B: there exists a sequence {Qn } (n ≥ 2) of algebraic polynomials Qn ∈ Πn such that, for each x ∈ [−1, 1], | D(ρ) (f (r) (x)) − Qn (x) | ≤ Cnβ (Δn (x))α+β . Applying Theorem 2.10.3 we obtain that condition B is equivalent to condition (i). Thus we have proved that (i) and (ii) are equivalent.

56

Chapter 2. The End Points Effect

Let us prove that (ii) and (iii) are equivalent. By applying r-times we obtain that condition (iii) is equivalent to condition A. From Theorem 2.10.4 (condition (i) and (ii) are equivalent) we obtain that condition A is equivalent to condition B: for each n ∈ N (n ≥ 2) there exist algebraic polynomials Tn,1 , Tn,2 ∈ Πn such that, for each x ∈ [−1, 1], | f (r+1) (x)(1 − x2 )i−1 − Tn,i (x) | ≤

α+β

C (Δn (x)) n2(ρ−1)−β

,

i = 1, 2.

If ρ > 1, then from Theorem 2.10.4 we obtain that condition B is equivalent to the following condition C: for each n ∈ N (n ≥ 2) there exist algebraic polynomials Hn,i ∈ Πn (i ∈ {1, 2, 3, 4}), such that, for each x ∈ [−1, 1], | f (r+1+i) (x)(1 − x2 )i−1 − Tn,i (x) | ≤

α+β

C (Δn (x)) n2(ρ−2)−β

,

i = 1, 2

and | f (r+1+i) (x)(1 − x2 )i−3 (1 − x2 )(i−2) − Tn,i (x) | ≤

α+β

C (Δn (x)) n2(ρ−2)−β

, i = 3, 4.

It can be proved that condition C is equivalent to the condition D: for each n ∈ N (n ≥ 2) there exist algebraic polynomials Ln,i ∈ Πn (i ∈ {1, 2, 3}), such that, for each x ∈ [−1, 1], | f (r+1+i) (x)(1 − x2 )i−1 − Ln,i (x) | ≤

α+β

C (Δn (x)) n2(ρ−2)−β

,

i = 1, 2, 3.

If ρ > 2, we repeat ρ − 2-times the arguments given above to obtain that condition D is equivalent to the following condition E: for each n ∈ N (n ≥ 2) there exist algebraic polynomials Sn,i ∈ Πn (i ∈ {0, 1, 2, . . . , ρ}), such that, for each x ∈ [−1, 1], | f (r+2ρ−i) (x)(1 − x2 )ρ−i − Sn,i (x) | ≤ C nβ (Δn (x))α+β ,

i = 0, 2, . . . , ρ.

From Theorem 2.10.3 we obtain that condition E is equivalent to condition (ii). Thus we have proved that conditions (ii) and (iii) are equivalent. 

2.11 Integral metrics In the works of Timan and Dzyadyk the best approximation by algebraic polynomials was well studied in the case of the uniform norm. Several authors considered that extension of the Timan-type estimates the spaces of integrable functions. The problem of characterization for some classes of functions was considered by Potapov and Lebed.

2.11. Integral metrics

57

In 1956 Potapov [287] extended the Timan theorem by considering functions with derivative of order r (r > 0, integer) is in Lip(p, α), p > 1, 0 < α ≤ 1. He also studied functions satisfying the condition  1/p d dx | f (r) (x + h) − f (r) (x) |p  ≤ M (f ) | h |α , (x − a)(b − x) c where a ≤ c < d ≤ b. In 1958 Lebed obtained a direct result. He considered the term Δn (x) as a varying weight. Theorem 2.11.1 (Lebed, √ [226]). Assume that p ≥ 1 and 1 − s − 1/p ≥ 0. If f ∈ C m [−1, 1] and ( 1 − x2 )s f (m) (x)p ≤ M , then there exists a sequence {Pn } (Pn ∈ Pn ) such that    f (x) − Pn (x)  M    (Δn (x))m−s  ≤ C(m) ns . p Denote by W (r) Hpw the class of functions given on the interval [−1, 1] and having an rth derivative f (r) whose pth power is integrable, and for which the inequality f (r) (x + h) − f (r) (x)Lp [−1,1−h] ≤ w(h),

0 < h < 1,

hods, where w is a fixed modulus of continuity. The class W (r) Aw p is defined analogously, but with the condition    f (r) (x + √1 − h2 − h√1 − x2 ) − f (r) (x)    √   ≤ C.   w(h 1 − x2 + h2 ) p (r+α)

(r+α)

(Ap respectively). For w(t) = tα we shall denote these classes by Hp (r) w The classes W Hp were introduced by Lebed and Potapov (see [290]). w They proved that W (r) H∞ = W (r) Aw ∞ (uniform norm). It is also obvious that the intersection of these classes is not empty, for 1 ≤ p < ∞. Potapov also used classes defined by two parameters. For 1 ≤ p < ∞, r ∈ N0 , (r) (r) ∈ Lp [−1, 1] if 0 ≤ β ≤ 1 and 0 < α ≤ 1, f ∈ Hp Aα β if f   1/p  f (r) (x√1 − h2 − h√1 − x2 ) − f (r) (x) p   √ ≤| h |α   dx  1 − x2 + | h |β −1 



1

in the case 0 < α < 1 and   1  (r)  f (λ(x, h)x − λ(h, x)) − 2f (r) (x) + f (r) (λ(x, h) + λ(h, x)) p   dx ≤| h |p √   2 + | h |β 1 − x −1 √ in the case α = 1, where λ(x, h) = x 1 − h2 . Here is a typical result.

58

Chapter 2. The End Points Effect (r)

Theorem 2.11.2 (Potapov, [289]). For a function f one has f ∈ Hp Aα β if and only if, for each n ≥ r + 2, there exists a polynomial Pn ∈ Pn , such that  p 1/p   C  √ f (x) − Pn (x)  ≤ r+α ,  ( 1 − x2 + 1/n)r+β  dx n −1



1

where the constant C does not depend on n or f . The paper of Potapov also contains analogous results when the Lebesgue measure is changed by the Chebyshev one. Some other results were presented in [290]. The following result follows from the works of Lebed and Potapov. Theorem 2.11.3 (Lebed-Potapov). For α ∈ (0, 1) and a function f , one has f ∈ A(r+α) if and only if for each n ≥ r there exists a polynomial Pn ∈ Pn , such that  p 1/p   C  √ f (x) − Pn (x)  ≤ r+α ,  ( 1 − x2 + 1/n)r+α  dx n −1



1

where the constant C does not depend on n or f . Taking into account (2.25), it was natural to look for weighted spaces. In this way some class of functions can be studied, but the original problems (characterization of classical Lipschitz spaces in terms of the best algebraic approximation or a characterization of a class of functions with a given rate for the best algebraic approximation in terms of the classical Lipschitz classes) was not solved. Since weighted approximation will not be discussed here in detail, we have included only a few remarks. (r+α)

The characterization of the class Hp was also considered by Motornyi in 1971. He verified that the quantity    f (x) − P (x)     λn (f ) = inf  P ∈P (Δn (x))α Lp (α)

(r+α)

(r+α)

are unbounded in the class Hp and established that classes Hp and Ap are different for 0 < α < 1 and coincide for α = 1. He also characterized some functions, but not in terms of approximation by polynomials (see Theorem 11 of [255]) on the whole interval. Oswald [277] extended some of the results of Motornyi to the case of moduli of smoothness of higher order. In 1978 DeVore [89] showed that we can not obtain a result similar to Theorem 2.7.7, if in (2.25) we replace the uniform norm by the Lp [−1, 1] norm, 1 ≤ p < ∞. That is, by considering    f (x) − p(x)     . Fn (f, r, α)p = inf  (2.50) Δr+α (x) p p ∈ Pn n

2.11. Integral metrics

59

In fact DeVore showed (with an incomplete proof) that for 0 < α < 1 and 1 ≤ p < ∞, ω1 (f, t)p = O(tα ) =⇒ Fn (f, 0, α)p = O(log n). Moreover, for each 0 < α < 1 there exists f ∈ Lp [−1, 1] such that ω1 (f, t)p = O(tα ) and Fn (f, 0, α)p ≥ C log n for infinitely many n. As we remarked above, Motornyi (α) proved that these quantities are not bounded when f varies on the class Hp . In 1972 Golischek presented a detailed study of this kind of weighted approximation [143]. For 1 ≤ p ≤ ∞ set  En(λ) (f )p = inf (max{1/n, 1 − x2 })−λ (f (x) − p(x)p p∈Pn

and consider the following question: under what conditions are the statements En(λ) (f )p = O(n−β ) and (max{1/n,

 1 − x2 })r−λ Pn(r) (x)p = O(nr−β )

(2.51) (2.52)

equivalent, where r ∈ N and β is a real number 0 < β < r? The answer is different for λ ≤ 0 and λ > 0. If λ ≤ 0, Golitschek proved that (2.51) and (2.52) are equivalent. For λ > 0 the situation is more complicated: if r > max{β, (λ + β)/2}, then (λ) (2.51) implies (2.52). Moreover, if we assume En (f )p = 0, then (2.52) implies (2.51). Golitschek also constructed a class of functions for which both assertions are equivalent. The following theorem generalizes some results of Motornii in [254]. Theorem 2.11.4 (Shalashova, [337]). Fix k ∈ N and p ∈ [1, ∞). Suppose f ∈ Lp [−1, 1] and ωk (f, t)p ≤ Ψ(t), where Ψ(t) is some positive function satisfying the conditions: 1) Ψ(t) does not decrease, 2) Ψ(λt) ≤ (λ + 1)k Ψ(t) for λ > 1. Then for any integer n > k, one can find an algebraic polynomial Pn of degree not greater than (4k + 2)n + k − 1 such that    f (x) − pn (x)  1/p    Ψ(Δn (x))  ≤ Ak [log(n + 1)] , p where Ak is a constant depending only on k. Since Ak does not depend on p, we arrive at the uniform estimate of Brudnyi by letting p tend to ∞ in the last inequality (for a bounded f ). If r = k + 1, and ω1 (f (k) , t) ≤ Ctα (0 < α ≤ 1), we obtain from the last result a theorem of Motornyi [254].

60

Chapter 2. The End Points Effect

Some authors have studied the best approximation of particular classes of functions. For instance, Nasibov considered the approximation by algebraic polynomials of functions of the form  1  x−t f (x) = ψ ϕ(t)dt (2.53) 2 −1 in the metric of Lp [−1, 1]. Theorem 2.11.5 (Nasibov, [266]). Let 1 ≤ p, r < ∞ and assume that ψ ∈ Lr [−1, 1] and ϕ ∈ Lp [−1, 1]. If f is defined by (2.53), then En (f )p ≤ 22−1/r ϕp En (ψ)r . Dynkin used a complex variable method (pseudo-analytical extension of functions) to obtain some results. For s > 0 and 1 ≤ p ≤ ∞ he gave a characterization of functions satisfying ∞ 1/p

1 p < ∞, En (f )p, s n k=1

where

 1/p   f (x) − p(x)    .  Δs (x)  dx −1 n

 En (f )p, s =

inf p ∈ Pn

1

Recall that for r ∈ N and 1 < p < ∞, Wpr [−1, 1] is the class of functions such that f (r−1) is absolutely continuous and f (r) ∈ Lp [−1, 1]. Oswald [277] considered the classes Wm of increasing functions ω such that ω ω(h) ≤ 2m ω(h/2) and Hp,m of functions in Lp such that ωm (f, t)p ≤ C(f )ω(t). Theorem 2.11.6 (Oswald, [277]). Fix m ∈ N and p ∈ [1, ∞). For each f ∈ Lp [a, b] and n ≥ m − 1,  b−a En (f )p ≤ C(m) ωm f, . n+1 p From the inequality ωm+r (f, t)p ≤ tr ωm (f (r) , t)p , it follows that, for f ∈ Wpr [a, b],  En (f )p ≤ C(m + r)

b−a n+1

r

 b−a ωm f (r) , . n+1 p

w . Given f ∈ Lp [a, b], 1 ≤ p < ∞, It can be used to characterize class Hp,m and m ∈ N, it is known that for any interval [c, d], [a, b] ⊂ (c, d), there exists an extension f ∗ of f to [c, d] such that,

ωm (f ∗ , h)p ≤ Cωm (f, h)p .

2.11. Integral metrics

61

If ω satisfies the condition 

H

hm h

ω(t) dt ≤ cω(t), tm+1

ω and En (f ∗ )p ≤ Cω(1/n) are equivalent. then the following conditions f ∈ Hp,m

Theorem 2.11.7 (Dynkin, [104]). Fix 1 < p < ∞ and r ∈ N. For a function f : [−1, 1] → R one has f ∈ Wpr [−1, 1] if and only if, for each k ∈ N0 , there exists P2k ∈ Π2k such that p/2  1 

∞ | f (x) − P2k (x) |2 dx < ∞. (Δ2k (x))2r −1 k=0

It was Operstein, in 1995 [275], who stated the theory as completed as in the uniform norm. Let ω√: R+ → R+ satisfy the condition ω(s + t) ≤ M (ω(s) + ω(t)) and set ρk (x) = 2−k 1 − x2 + 2−2k . We use the customary notation for the mixed norm Ak (·)lp (Lp ) = {Ak (·)Lp }k lp . That is  Ak (·)lp (Lp ) =

∞ 

k=1

1/p

1

−1

| Ak (x) |p dx

= {Ak (·)Lp }k lp .

Theorem 2.11.8 (Operstein, [275]). Fix p ∈ [1, ∞] and r ∈ N. There exists a constant C = C(p, r) such that, for each f ∈ Lp [−1, 1] and k ∈ N0 , there exists an algebraic polynomial {Pk } of degree at most 2k + r − 2 such that      f − Pk   ωr (f, 2−k )     .   ≤ C   ω(ρk )  ω(2−k ) lp lp (Lp ) Brudnyi’s Theorem 2.5.3 follows from this one by setting ω(t) = ωr (f, t)p and p = ∞. Theorem 2.11.9 ([275]). Let f be a function defined on [−1, 1]. If there exists a sequence {Pk } of algebraic polynomials of degree at most 2k − 1 such that    f − Pk    ≤ 1,  ω(ρk )  lp (Lp ) then for every r ∈ N,  ωr (f, t)p ≤ C t

r t

1



ω(u) ur

q

du u

where the constant C depends only on r and p.

1/q ,

1 1 + = 1, p q

62

Chapter 2. The End Points Effect

When p = ∞ we obtain the Timan inverse result. With these theorems one has the characterization of Lip(α, p) spaces. Theorem 2.11.10 (Operstein, [275]). A function f : [−1, 1] → R belongs to Lip(α, p) if and only if there exists a sequence {Pk } of algebraic polynomials of degree at most 2k (k = 0, 1, . . . ) such that (f − Pk ) min{1, t/ρk }s lp (Lp ) = O(tα ),

0 < α < s.

The idea of using min{1, t/ρk } for a characterization of Lip(α, p) appears in [89], where it is proved that for each function f ∈ Lip(α, p), 0 < α < 1, there exists a polynomial Pk such that (f − Pk ) min{1, t/ρk }lp (Lp ) = O(tα ). As we remarked above, Motornyi and DeVore showed that the direct analogue ((f − Pn )ρ−α n }Lp ≤ C does not characterize Lip(α, p) when p < ∞.

2.12 Lp , 0 < p < 1 The behavior os the best approximation in Lp space, for 0 < p < 1 is not the same as in the case p ≥ 1. For instance, the difference f (x) − Pn (x) (where Pn is a polynomial of the best approximation) must not oscillate at least at n + 1 point. For studies concerning this problem see [160], [402], [403], [404] and [405]. For 0 < p < 1 the functional 

1/p

b

| f (x) | dx

f p =

p

a

is not a norm, but the notation f p is used in this case for the sake of convenience. Some smoothing processes which are usually applied in approximation theory do not work well in Lp spaces (0 < p < 1). Even more, the common definition of Sobolev spaces gives place to spaces with a trivial dual (see [279]). Thus, the ideas associated to K-functionals can not be used. There are also differences with the classical spaces related with the connection between smoothness and the existence of derivatives. In [214] Kortov studied this last topic. In 1975 Storozhenko, Krotov and Oswald (Osval’d) presented direct and converse results for trigonometric approximation in the space of periodic functions Lp [0, 2π], for 0 < p < 1 [356]. The extension of the classical theory to this setting was motivated by some problems related with embedding theorems (see [352]). In [357] Storonzenko and Oswald presented estimates with the second-order modulus. Theorem 2.12.1. If 0 < p < 1 and f ∈ Lp [0, 2π], then  π ∗ , En (f )p ≤ Cp ω f, n+1 p

2.12. Lp , 0 < p < 1

63

where En∗ (f )p = inf

Tn ∈Tn

f − Tn p .

Moreover, for n = 0, 1, . . . ,  ω f,

1 n+1

p

⎞1/p ⎛ n Cp ⎝

≤ (j + 1)p−1 (Ej (f )p )p ⎠ . n + 1 j=0

(2.54)

Similar results were obtained in the same year by V.I. Ivanov [161], but he used moduli of smoothness of higher order: ⎛ ⎞1/p  n 1 Cp,k ⎝

ωk f, ≤ (j + 1)kp−1 (Ej (f )p )p ⎠ . nk n p j=0 In [356] and [161] a Bernstein inequality was proved for spaces Lp [0, 2π], 0 < p < 1, in the form T (r)p ≤ C(p)nr T p. Other proofs were given by Ivanov [162], Oswald [276], Nevai [269] and Runovskii [320]. The best result was presented by Arestov. Theorem 2.12.2 (Arestov, [2]). For 0 < p < 1, n, r ∈ N and Tn ∈ Tn one has Tn(r)p ≤ nr Tn p . In [320] and [321] Runovskii constructed some linear polynomial operators and obtained direct results in Lp [0, 2π] (0 < p < 1) in the periodical case. For 0 < p < ∞ and μ ≥ −1/p, Khodak considered the spaces Lp,μ [−1, 1] of functions f for which 1/p  1  f p,μ = | f (x)( 1 − x2 )μ |p < ∞. −1

if A function f ∈   1/p  π  f (cos(γ + t)) − f (cos γ) p   (sin γ)1+μp dγ ≤ C | sin t |α .   (sin γ+ | sin t |)β 0 Aα,β p,μ

α,β

A function f ∈ Ap,μ if  1/p  π   f (cos(γ + t1 )) + f (cos(γ + t2 )) − 2f (cos(γ + (t1 + t2 )/2)) p   (sinγ)1+μp dγ   (sinγ+ | sint |)β 0 ≤ C | sint |α , where t =| t1 | + | t2 |. Here β is a real number, for p ≥ 1 we consider that 0 < α ≤ 1 and, for 0 < p < 1, 0 < α ≤ 1/p.

64

Chapter 2. The End Points Effect

For p ≥ 1, and t1 = −t2 these spaces coincide with the one studied by Lebed and Potapov. Theorem 2.12.3 (Khodak, [189]). Let f ∈ Lp,μ [−1, 1], 0 < p < 1, μ ≥ −1/p, 0 < α < 2, −2/p − 2 − μ + α < −β < α − 1/p − μ. α,β

In order that f ∈ Aα,β p,μ for 0 < α < 1 or f ∈ Ap,μ for 0 < α < 2, it is necessary and sufficient that there exists a sequence {Pn }, Pn ∈ Pn such that   −β   1 C   1 − x2 + [f (x) − Pn (x)]  ≤ α ,   n n p,μ

where the constant C does not depend on f and n. As Ditzian showed we can not extend the results related with simultaneous approximation to the case 0 < p < 1. Theorem 2.12.4 (Ditzian, [97]). For each 0 < p < 1 there exists a function f ∈ A.C.[−1, 1] for which we can not find a sequence {pn }, pn ∈ Pn such that f − pn p ≤ Cω2 (f, 1/n)p

and

f  − pn p ≤ Cω(f  , 1/n)p .

The same assertion holds if we replace the usual moduli by the Ditzian-Totik one.

2.13 The Whitney theorem Another form for the direct results in approximation by algebraic polynomials is due to Whitney. Theorem 2.13.1 (Whitney, [407] and [408]). For any n ∈ N there exists a constant W∞ (n) such that, for every bounded function f : [a, b] → R there exists a polynomial Pn−1 (f ) ∈ Pn−1 satisfying  b−a . f − Pn−1 (f ) ≤ W∞ (n) ωn f, n In fact, this was proved by Burkill in 1952 [43] for n = 1, 2, who also conjectured that the inequality holds for n ≥ 3. In 1957 Whitney verified the conjecture for continuous functions and in 1959 for bounded functions. The proof of Whitney, as the one due to Burkill, used the polynomial P which interpolates f over a uniform net   k k P =f , (k = 0, 1, . . . , n − 1). n−1 n−1

2.13. The Whitney theorem

65

If we consider the polynomial Qn−1 (f ) which interpolates f at a uniform net of node, then we can also consider the inequalities  b−a  f − Qn−1 (f ) ≤ W∞ (n) ωn f, . n In 1964 Brudnyi found a new proof of the Whitney theorem. He used some smoothing of the function by means of linear combinations of Steklov-type functions. With the new method of proof he was able to extend the result to Lp spaces, with 1 ≤ p < ∞. In 1977 Storozhenko extended the Whitney theorem for algebraic approximation to Lp [a, b] spaces, for 0 < p < 1 (see also [358]). Theorem 2.13.2 (Brudnyi, [39] and Storozhenko, [353]). Suppose 0 < p < ∞, f ∈ Lp (a, b) and n is an arbitrary natural number, then  b−a , En−1 (f )p ≤ Wp (n) ωn f, n p where Wp (n) depends not on f . Another proof was presented in [355] by Storozhenko and Kryakin. In [354] Storozhenko presented the inequality: for 0 < p < 1, f ∈ Lp [−1, 1], k ∈ N and n ≥ k − 1,  1 En (f )p ≤ Cp,k ωk f, . n+1 p A similar inequality appeared in [342] but only for the first modulus. Another proof was given by Khodak in [190]. The proof of Whitney can not be used as the estimate of the constants. Whitney proved that 1 ≤ W∞ (n) 2 and found some bounds for some values of n. For instance 1 ≤ W∞ (1) ≤ 2,

1 ≤ W∞ (2) ≤ 2.

The Whitney theorem has been studied by several Bulgarian mathematicians. In 1982 Sendov conjectured that W∞ (n) ≤ 1 [331]. This motivated several papers, shown in the table on top of the next page. The inequality W∞ (n) ≤ 1 has been verified only for a few values of n: Whitney n = 3 [407], Kryakin n = 4 [219] and Zhelnov k = 5, 6, 7, 8, [416].  (n) ≤ 5ωn (f, 1/n). In [221] Kryakin and Takev proved that W∞ Tunc [391] considered Whitney-type theorems in the form   r b−a b−a Ek+r+1 (f, [a, b])∞ ≤ W∞ (k, r) , [a, b] ωk f, . k k ∞ He found upper bounds for W (k, 2), W (1, r) and W (2, r).

66

Chapter 2. The End Points Effect

Year

Author

Reference

Estimate

1952

Burkill

[43]

W∞ (2) = 1/2,

1964

Brudnyi

[39]

W∞ (n) ≤ Cn2n ,

1985

Ivanov-Takev

[174]

W∞ (n) ≤ C(n ln n),

1985

Binev

[31]

W∞ (n) ≤ C n,

1985

Sendov

[332]

W∞ (n) ≤ C,

1986

Sendov

[333]

W∞ (n) ≤ 6,

1985

Sendov-Takev

[335]

W1 ≤ 30,

1989–90

Kryakin

[215], [216]

W∞ (n) ≤ 3,

1989

Sendov-Popov

[334]

W∞ (n) ≤ 3,

1990

Kryakin

[216]

Wp (n) ≤ 11,

1992

Kryakin-Kovalenko

[220]

W1 ≤ 6.4,

1992

Kryakin-Kovalenko

[220]

Wp ≤ 9,

1995

Kryakin

[217], [218]

W∞ (n) ≤ 2.

2002

Gilewicz-KryakinShevchuk

[142]

W∞ (n) ≤ 2 + e−2 .

2.14 Other classes of functions Bernstein [30] characterized C ∞ [a, b] as follows: f ∈ C ∞ [a, b] if and only if each k ∈ N, lim nk En (f ) = 0. n→∞

Some subclasses of functions of C ∞ [a, b] has been studied by Brudnyi-Gopengauz [41] Babenko [4] and Motornyi [256].

Chapter 3

Looking for New Moduli Different authors have tried to used other forms of measuring the smoothness of functions. In the first section of this chapter we present some of the ideas associated to the works of Potapov. In the second section we analyze the circle of ideas developed by Butzer and his collaborators.

3.1 The works of Potapov Potapov began to consider the approximation by algebraic polynomials in Lp spaces in 1956 [287], where he follows Timan’s ideas. In 1960 and 1961 he obtained results in which the usual translation was modified ([289] and [290]). Let Lp,α,β [−1, 1] be the space of all functions f for which f p,α,β = f (x)(1 − x)α (1 + x)β p < ∞ and En (f )p,α,β be the best approximation by algebraic polynomials in this space. That is En,α,β (f ) = inf f − Pn p,α,β . Pn ∈Pn

When α = β we simply write Lp,α [−1, 1] and En (f )p,α . Let us denote by ω(f, t)p,α,β the usual modulus of continuity of f in the metric of Lp,α,β [−1, 1]. That is ω(f, t)p,α,β = sup f (x + h) − f (x)p,α,β , |h| ≤ t

with the usual restriction relative to the interval (f (x+h)−f (x) = 0, if x+h > 1). In 2000 Potapov proved that in Lp,α,β [−1, 1] the usual Lipschitz classes can not be characterized by the best approximation in the same form as in the case of trigonometric approximation [300].

J. Bustamante, Algebraic Approximation: A Guide to Past and Current Solutions, Frontiers in Mathematics, DOI 10.1007/978-3-0348-0194-2_3, © Springer Basel AG 2012

67

68

Chapter 3. Looking for New Moduli

Let us set G(f, x, t) =

  1 [f (x cos t + 1 − x2 sin t) + f (x cos t − 1 − x2 sin t)] 2

(3.1)

and define ω ! (f, t)p,α,β = sup f (x) − G(f, x, h)p,α,β . |h| ≤t

Theorem 3.1.1 (Potapov, [289]). Fix p ∈ [1, ∞], α = β = −1/(2p) and γ ∈ (0, 1). For f ∈ Lp,α,β [−1, 1] the following assertions are equivalent: (i) There exists a constant M such that ω ! (f, t)p,α,β ≤ M tγ . (ii) There exists a constant K such that, for all n ∈ N, En (f )p,α,β ≤ K/nγ . For the un-weighted case (α = β = 0) and p = 2, Zhidkov obtained another characterization. Define ω " (f, t)p,α,β = sup f (x) − H(f, x, h)p,α,β , |h| ≤t

where H(f, x, h) =

1 π



 dy f (x cos t + y sin t 1 − x2 )  . 1 − y2 −1 1

(3.2)

Theorem 3.1.2 (Zhidkov, [417]). If γ ∈ (0, 1), for a function f ∈ L2 [−1, 1] there exists a constant M such that ω " (f, t)2,0,0 ≤ M tγ if and only if there exists a constant K such that, for all n ∈ N, En (f )2,0,0 ≤ K/nγ . Let us recall another result of Zhidkov. Theorem 3.1.3 ([417]). For f ∈ L2 [−1, 1] one has En (f )2 ≤ C/ns+γ if and only if 

1

−1



ds fh (x) ds f (x) − dxs dxs

2

1/2 2 s

(1 − x ) dx

≤ C hγ ,

where h > 0, n > s, 0 < γ < 1 and   1 π fh (x) = f (x cos h + 1 − x2 sin h cos t)dt. π 0 In [292] Potapov considered the problem of characterizing all functions f ∈ Lp,α,β [−1, 1] for which there exists a sequence of algebraic polynomials satisfying  ρ1  ρ2     1 1 (f (x) − Pn (x)) 1 − x + 1  1+x+ 2 ≤ C r+γ .    2 n n n p,α,β The case α = β ≥ −1/(2p) and ρ1 = ρ2 has been studied previously by him in [291]. The results are given in terms of a generalized translation.

3.1. The works of Potapov

69

For f : [−1, 1] → R, consider it a Fourier-Jacobi series f (x) ∼

∞

(α,β)

ak Pk

(x)

k=0 (α,β)

}) is the sequence of Jacobi polynomials. That is, they are the orwhere {Pk thogonal polynomials in [−1, 1] with respect to weight (1 − x)α (1 + x)β , with the (α,β) normalization Pk (1) = 1. Let us consider the associated series Th (f, x, α, β) =

∞

(α,β)

ak Pk

(α,β)

(x)Pk

(h).

(3.3)

k=0

Assume that, for each h ∈ [−1, 1], there exists a function gh such that (3.3) holds if the Fourier-Jacobi series of gh holds, then we consider that (3.3) is the FourierJacobi series of f with generalized translation x + h. The functions were called the generalized translation by L¨ ofstr¨ om and Peetre in [238]. Now the generalized modulus is defined by ω(f, t, α, β)p,α,β = sup f (x) − Tcos s (f, x, α, β)p,α,β .

(3.4)

|s| ≤ t

It is not a simple task to find a simple expression for the generalized translation Th (f, x, α, β). In the case α = β = −1/2, the Fourier-Jacobi polynomials are just the (−1/2,−1/2) (x) = Tn (x) = cos(n arccos x)). It can be Chebyshev polynomials: Pn proved that Tcos t (f, x, −1/2, −1/2) = G(f, x, t), where G(f, x, t) is the function defined (3.1). In this case the direct and converse results were recalled in Theorem 3.1.1. In the case α = β = 0, the Fourier-Jacobi polynomials are the Legendre polynomials and the translation has the form Tcos t (f, x, 0, 0) = H(f, x, t), where H(f, x, t) is defined by (3.2). In this case the direct and converse results were recalled in Theorem 3.1.2 (in L2 spaces). Recall that the Legendre polynomial Pn (x) of degree n is defined by Pn (x) =

(−1)n dn (1 − x2 )n , 2n n! dxn

(x ∈ [−1, 1],

n ∈ N0 ).

(3.5)

70

Chapter 3. Looking for New Moduli

For α = β > −1/2, the Fourier-Jacobi polynomials are the Gegenbauer polynomials and the translation has the form  1  1 Tcos t (f, x, α, α) = f (x cos t + y 1 − x2 sin t)(1 − y 2 )α−1/2 dy, (3.6) γ(α) −1 1 where γ(α) = −1 (1 − y 2 )α−1/2 dy. In this case the direct and converse results were given by Rafalson [311] and Pawelke [278]. Rafalson extended the theorem of Zhidkov [417] for the case α > 0. Zhidkov and Rafalson only considered approximation in L2 spaces. For α > β = −1/2, the translation is given by  1 1 f (Φ1 (x, t, y))Θ1 (y)dy, (3.7) Tcos t (f, x, α, β) = γ1 (α, β) −1 where Φ1 (x, t, y) = x cos t + y



1 − x2 sin t − (1 − y 2 )(1 − x) sin2 (t/2),

Θ1 (y) = (1 − y 2 )α−1/2 . and γ1 (α, β) is chosen from the condition Tcos t (1, x, α, β) = 1. For α > β > −1/2, the Fourier-Jacobi polynomials are the Jacobi polynomials and the translation has the form  1 1 1 Tcos t (f, x, α, β) = f (Φ2 (x, t, r, y))Θ2 (r, y)dydr, (3.8) γ2 (α, β) 0 −1 where Φ2 (x, t, r, y) = x cos t + ry



1 − x2 sin t − (1 − r2 )(1 − x) sin2 (t/2),

Θ2 (r, y) = (1 − r2 )α−β−1 r2β+1 (1 − y 2 )α−1/2 and γ2 (α, β) is chosen from the condition Tcos t (1, x, α, β) = 1. In this case the direct and converse results were given by Potapov in [292]. The case α = 0 and β > −1, was studied by Potapov, Fedorov and Fraguela in [309] and [308]. They wrote the generalized translation as Ttβ (f, x)

=

where | t |< π, and

1 π cos2β (t/2)





π

f (cos s) 0

1 + cos s 1+x

β

 cos s = x cos t + cos u sin t 1 − x2 ,

cos(2βr)du

0≤r≤π

√ √ 1 + x cos(t/2) + cos u 1 − x sin(t/2)  cos r = . √ 1 + x cos t + cos u 1 − x2 sin t

3.1. The works of Potapov

71

With this translation the modulus is defined by Ω(f, δ)p,β = sup [Ttβ (f, x) − f (x)](1 + x)β p . |t|≤δ

Theorem 3.1.4 (Potapov-Fedorov, [308]). Suppose that β > −1/2 and 1 ≤ p ≤ ∞. There exists a positive constant C1 and C2 such that, for every f ∈ Lp,0,β [−1, 1] and n ∈ N,  n 1 C2

≤ 2 kEk (f )p,0,β . C1 En (f )p,0,β ≤ Ω f, n n p,β k=1

For given ν and μ, assume that the translation H(f, t, ν, μ) is defined by (3.1) when ν = μ = −1/2, by (3.6) when ν = μ > −1/2, by (3.7) when ν > μ = −1/2 and by (3.8) when ν > μ > −1/2. With this selection define the modulus ω(f, δ, μ, ν)p,α,β = sup f (x) − H(f, t, ν, μ)p,α,β . |t|≤δ

Theorem 3.1.5 (Potapov, [297]). Fix p ∈ [1, ∞] and α ≥ β ≥ −1/(2p). Assume that ν and μ are chosen following the rules: μ = ν = −1/2, μ = −1/2, ν > α − 1/2 + 1/(2p), ν = μ > α − 1/2 + 1/(2p), μ > β − 1/2 + 1/(2p), ν > μ + α − β,

if if if

α = β = −1/(2p), α > β = −1/(2p), α = β > −1/(2p),

if

α > β > −1/(2p).

There exist positive constants C1 and C2 such that, for all f ∈ Lp,α,β ,  n 1 C2

C1 En (f )p, α, β ≤ ω f, , ν, μ ≤ kEk (f )p, α, β . n n p, α, β k=1

Some other results concerning Jacobi weights were given by Potapov in [296]. Some extensions to moduli of higher order were presented by Tankaeva in [370] and by Potatov and Kazimirov in [310]. Other results for Jacobi weights were obtained by Potapov, Berisha and Berisha in [307]. In 1999 Potapov provided another modulus using a non-symmetric generalized operator of translation. He considered the expression given in (3.3) as the (α,β) symmetrical case and replace the term Pk (h) by ϕk (h). The new formula Th (f, x, α, β) =

∞

k=0

is called non symmetric.

(α,β)

ak Pk

(x)ϕk (h),

(3.9)

72

Chapter 3. Looking for New Moduli

For the case α = β, he considered the translation  π 1 Tt (f, x) = f (Φ3 (x, t, s))Θ3 (x, t, s)ds, π(1 − x2 ) 0 

where Φ3 (x, t, s) = x cos t + cos s sin t

1 − x2

Θ3 (x, t, s) = 1 − (Φ3 (x, t, s))2 − 2 sin2 t sin2 s + 4(1 − x2 ) sin2 t sin4 s and the modulus ω ! (f, δ)p,α = sup Tt (f, x) − f (x)p,α . |t| ≤ δ

Theorem 3.1.6 (Potapov, [298]). Fix p, α and r such that r ∈ (0, 2) and α ∈ (1/2, 1], if α ∈ (1 − 1/(2p), 3/2 − 1/(2p)), if α ∈ [1, 3/2), if

p = 1, 1 < p < ∞, p = ∞.

For a function f ∈ Lp,α [−1, 1] the following assertions are equivalent: En (f )p,α ≤ C(f ) n−r , ω ! (f, δ)p,α ≤ C(f )δ r .

(i) (ii)

Extension to moduli of smoothness of order r were given by Potapov and Berisha in [304] (see also [305]). In the case α = β + 1, the translation is defined by  4 1 dz Ty (f, x) = f (R)ψ(x, y, z) √ , π −1 1 − z2 where

√ cos(u + μ − u1 )(1 − R) 1 − R2 √ ψ(x, y, z) = , (1 + y)2 (1 − x) 1 − x2   R = xy + z 1 − x2 1 − y 2 ,  sin u1 = 1 − z 2 , cos u1 = z, 

√ 1 − y 2 x + yz 1 − x2 √ , cos u = 1 − R2  √ z(1 − xy) − 1 − x2 1 − y 2 , cos μ = 1−R −

√ √ 1 − x2 1 − z 2 √ sin u = , 1 − R2 √ 1 − z 2 (y − x) . sin μ = 1−R

Now define ω " (f, δ)p,α,β = sup Tcos t (f, x) − f (x)p,α,β . |t| ≤ δ

3.2. Butzer and the method of Fourier transforms

73

Theorem 3.1.7 (Potapov, [299]). Fix p ∈ [1, ∞] and assume α ∈ (0, 1/2], p = 1, α ∈ (1/2 − 1/(2p), 1 − 1/(2p)), 1 < p < ∞, α ∈ [1/2, 1), p = ∞. There exist positive constants C1 and C2 such that, for all f ∈ Lp,α+1,α , C1 En (f )p, α+1, α ≤ ω " (f, 1/n)p, α+1, α ≤

n C2

kEk (f )p, α+1, α . n2 k=1

3.2 Butzer and the method of Fourier transforms The methods of Fourier transforms can be used to prove of the classical assertions for trigonometric approximation into the Jacobi-weighted frame. Several authors are related to this topic. Ganser [130] introduced the modulus of continuity in the Jacobi frame. The works began with Bavinck ([19] and [20]) and Scherer and Wagner [330] in 1972. Butzer and Stens ([55], [56] and [57]) introduced the Chebyshev transform method and Butzer, Stens and Wehrens ([58], [59], [60] and [350]) the Legendre transform method. Finally, the Jacobi transform method was presented in [60]. Some results concerning Gegenbauer-weights were given by L¨ofstr¨ om [237]. One of the disadvantages of the Jacobi transform method is that derivatives and Lipschitz classes are defined in terms of a generalized translation. Let us present some ideas taken from [55]. As usual, C[−1, 1] denotes the set of all continuous real-valued functions f defined on [−1, 1] with the sup norm. Let Lpw , 1 ≤ p < ∞, be the set of all measurable real-valued f on [−1, 1] for which the norm  f p =

1 π



1

−1

1/p | f (u) | w(u)du p

√ w(x) = 1/ 1 − x2 , is finite. Below, X stands for one of the Banach spaces C[−1, 1] or Lpw . For f ∈ X, the kth Chebyshev-Fourier coefficient is defined by  1 1 du f (u)Tk (u) √ T[f ](k) = [f ]∧ (k) = π −1 1 − u2

(3.10)

where Tk (x) = cos(k arccos x) (x ∈ [−1, 1]), is a Chebyshev polynomial of degree k. The classical translation of a function f (x) by h, namely f (x + h), is replaced by       f xh + (1 − x2 )(1 − h2 ) + f xh − (1 − x2 )(1 − h2 ) (3.11) (τh f )(x) = 2

74

Chapter 3. Looking for New Moduli

(x, h ∈ [−1, 1]). This translation has the advantage that it is an operator from X into itself and satisfies limh→1− τh f −f X = 0. Thus one can define the Chebyshev derivative as the function g ∈ X for which     f − τh f   lim − g  = 0, h→1−  1 − h X whenever such a function exists, and then we write D1 f = g. Derivatives Dr of higher order r = 2, 3, . . . are defined iteratively. r The set of all f ∈ X for which Dr f exists is denoted by WX . It was proved r in [56] that, for f ∈ X, one has f ∈ WX if and only if there exists g ∈ X such that (−k 2 )r T[f ](k) = T[g](k).

(3.12)

In this case Dr f = g − g ∧ (0). If we define the convolution product of f ∈ L1w and g ∈ X by  1 1 (f ∗ g)(x) = (τx f )(u)g(u)w(u)du, (3.13) 2 −1 then f ∗ g ∈ X, and its Chebyshev transform, satisfies T[f ∗ g](k) = T[f ](k)T[g](k).

(3.14)

The (right) difference of f ∈ X of order r ∈ N with respect to the increment h ∈ [−1, 1] is defined by 1

(Δh f )(x) = (τh f )(x) − f (x), r

1

(Δh f )(x) = (Δh (Δh

r−1

f ))(x).

With the notions presented above, the modulus of continuity and Lipschitz class are introduced as follows: r

ωrT (f, t) = sup (Δh f )X ,

(t ∈ [−1, 1])

(3.15)

t≤h≤1

and LipTr (α, X) = {f ∈ X : ωrT (f, t) = O((1 − t)α )}. There are relations between these notions and the usual moduli of continuity. If X = Lpw ,we denote by X2π the Lp space of 2π-periodic functions. Proposition 3.2.1. For f ∈ X, F ∈ X2π , η ∈ [−1, 1], δ > 0, α > 0 and r ∈ N, one has (i) ωrT (f, η) = ω2r (f ◦ cos, arccos η), (ii) f belongs to LipTr (X; α) if and only if f ◦ cos belongs to Lip2r (X2π ; 2α), (iii) if F is even, then F ∈ Lip2r (X2π ; 2α) if and only if F ◦ arccos ∈ LipTr (X; α).

3.2. Butzer and the method of Fourier transforms

75

In this setting Butzer and Stens presented an analogue of Theorem 1.2.1 for the best algebraic approximation. Theorem 3.2.2. Fix r, r1 , r2 ∈ N and 0 < α < 1. For a function f ∈ X the following assertions are equivalent: (i) En (f ; X) = O(n−2(r+α) ), (n → ∞), (ii) ω1L (Dr f ; δ) = O((1 − δ)α ), (δ → 1−), (iii) Dr1 p∗n (f )X = O(n−2(r+α−r1 ) ), (r1 > r + α, n → ∞), r2 (iv) f ∈ WW , Dr2 f − Dr2 p∗n (f )X = O(n−2(r+α−r2 ) ), (r2 < r + α, n → ∞). Proof. The method of proof follows the general approach of Theorem 1.4.1. Take Mn = Pn , ρ = r1 , σ = r1 and s = r + α. Moreover, set Y = W r1 , with seminorm | g |Y = Dr1 gX , and Z = W r2 , with seminorm | h |Z = Dr2 hX . It can be proved that Dr2 is a closed operator (see [56], Corollary 4). Hence, in view of the closed graph theorem, Z becomes a Banach space under the norm hZ = hX + Dr2 hX . It is known that, for each m ∈ N, there exist positive constants D1 = D1 (m) and D2 = D2 (m) such that, for f ∈ X and t ∈ (0, π] T m T (f, cos t) ≤ K(f, t2m , X, WX ) ≤ D2 ω m (f, cos t). D 1 ωm

Taking into account that, for all n ∈ N, Mn ⊂ Y ⊂ X, Mn ⊂ Z ⊂ X and that, for f ∈ X an element of the best approximation always exists in Mn , we only need to verify the Jackson- and Bernstein-type inequalities given in (1.17) and (1.18) hold. The Bernstein-type inequality follows from the classical Bernstein inequality for trigonometric polynomials. In fact, if n, m ∈ N and Pn ∈ Pn , then Dm Pn X = (Pn ◦ cos)(2m) ◦ arccos X ≤ (2n)2m Pn ◦ cos X,2π , where the last norm in computed on the interval [0, 2π]. Now, let us consider the Jackson-type inequality. Let Kn be the Fej´er-Korovkin operator (the formal definition is given in the section devoted to integral operators). Set Kn0 = I, Kn1 = Kn and, Knj = Kn1 (Knj−1 ) (j ∈ N). It can be proved that, for each j ∈ N and f ∈ X, Knj (f ) ∈ Pn . Set  r

r Knj . (−1)j+1 Ur,n = j j=1 Since Ur,n (f ) − f = (−1)r−1 (Kn − I)r (f ),

76

Chapter 3. Looking for New Moduli

then (see Corollary 5.5.7 below) Ur,n (f ) − f X = Kn ((Kn − I)r−1 (f )) − (Kn − I)r−1 (f )X  2   π π ≤ 1+ √ ω1T (Kn − I)r−1 (f ), cos 1 − cos . n+2 2 r In particular, there exits a constant C1 such that, if f ∈ WX , then C1 D1 ((Kn − I)r−1 (f ))X . n2 This yields the Jackson-type inequality in the case r = 1. If r > 1, taking into account that Ur,n (f ) − f X ≤

D1 ((Kn − I)r−1 (f )) = (Kn − I)r−1 (D1 (f )) and using the arguments given above we obtain a constant C2 such that C1 C2 Ur,n (f ) − f X ≤ 2 (Kn − I)r−1 (D1 (f ))X ≤ 4 D2 ((Kn − I)r−2 (f ))X . n n This yields the Jackson-type inequality in the case r = 2. By repeating this process we obtain the general assertion.  If we compare these results with the ones we presented above, we notice several facts. Fuksman worked with the classical derivative concept and did not include assertions concerning higher-order moduli. Asadov [3] and Khalilova [187] followed a similar approach, however only for functions which are quadratically integrable with respect to a weight. Dzafarov [105] considered continuous functions and used a different notion of derivative. Finally, Bavinck [19] examined spaces with weight (1 − x2 )β (1 − x2 )γ for certain values of β and γ, but he did not characterize the assertion En (f ) = O(n−2r ), r ∈ N. Other results related with the work of Butzer will be presented in the section devoted to Ditzian and Totik.

3.3 The τ modulus of Ivanov In order to obtain characterizations for the second interpretation after (1.11), Ivanov used the τ modulus ([163] and [166]). Given an arbitrary positive function δ and a non-negative continuous function w, define τk (f, w, δ)r, p, [a, b] = w(·)ωk (f, ·, δ(·))r p,[a,b] where

 ωk (f, x, δ(x))r =

1 2δ(x)



δ(x)

−δ(x)

 k  Δ f (x)r dv v

1/r .

In order to simplify we omit the index [a, b]. Moreover, when w ≡ 1 we omit w in the notation.

3.3. The τ modulus of Ivanov

77

Another τ modulus is defined by τk (f, t)∗p = ωk (f, ·, t)r p,[a,b] where     kt kt ∩ [a, b] . ωk (f, x, t) = sup | Δkh f (x) | : t, t + kh ∈ x − , x + 2 2 Several properties of these moduli, as well as its connection with ωk (f, t)p and τk∗ (f, t)p , were given in [163] and proved in [166]. Theorem 3.3.1. If 1 ≤ r, p, f, g ∈ Lmax{r,p} [a, b] and α ∈ R, then (i) (ii) (iii) (iv) (v)

τk (f + g, w, δ)r, p ≤ τk (f, w, δ)r, p + τk (g, w, δ)r, p , τk (αf, w, δ)r, p = | α | τk (f, w, δ)r, p , τk (f, w1 , δ)r, p ≤ τk (f, w2 , δ)r, p , 0 ≤ w1 ≤ w2 , τk (f, w, δ)r, p1 ≤ (b − a)1/p1 −1/p2 τk (f, w, δ)r, p2 , 1 ≤ p1 ≤ p2 , τk (f, w, δ)r1 , p ≤ τk (f, w, δ)r2 , p , 1 ≤ r1 ≤ r2 .

Theorem 3.3.2. For p, r, s ≥ 1, d > 0, n ∈ N and α ≥ 1, (i) (ii) (iii) (iv) (v) (vi)

τ1 (f, n d)1, p ≤ n τ1 (f, d)1, p , τ1 (f, α d)1, p ≤ (3 + [α]) τ1 (f, d)1, p , τk (f, d)r, p ≤ C(k) τk−1 (f  , d, d)s, p , , k ≥ 2, f  ∈ Lmax{p,s} , τk (f, d)r, p ≤ ωk (f, d)p ≤ C(k)τk (f, d)r, p , r ∈ [1, p], τ1 (f, d)∞, p ≤ τ1∗ (f, d)p ≤ 2 τ1 (f, d)∞, p , f ∈ L∞ , τk (f, d)∞, p ≤ C(k) τ1∗ (f, d)p , f ∈ L∞ , k ≥ 2.

Theorem 3.3.3. Suppose that the weight w√ satisfies the following condition: for every x, t ∈ [−1, 1] for which | x − t |≤ λ(d 1 − x2 + d2 ), w(x) ≤ C(λ) w(t).

(3.16)

For 1 ≤ p, r, s ≤ ∞, d ≤ 1 and f ∈ Lmax{p,r} (or f  ∈ Lmax{p,r} , or f (k) ∈ Lp ), then τk (f, w, Δ(d))r, p ≤ C(k) wf p , r ≤ p, τk (f, w, Δ(d))s, p ≤ C(k) τk−1 (f  , wΔ(d), Δ((4k + 2)d)r, p , k ≥ 2, τk (f, w, Δ(d))r, p ≤ C(k) Δk (d)f (k) p , k ≥ 1, τ1 (f, w, AΔ(d))r, p ≤ C(A)τ1 (f, w, Δ(d))r, p , d ≤ (2A)−1 , A ≥ 1, τk (f, w, Δ(d))r, p ≤ τk (f, w, Δ(d))s, p ≤ C(k) τk (f, w, Δ(d))r, p , 1 ≤ r ≤ s ≤ p. √ The weight w(x) = (d 1 − x2 + d2 )μ (μ real) satisfies (3.16) with a constant C(λ) = (4λ + 2)|μ| and w(x) ≡ √ 1 also satisfies (3.16). Let us present direct and converse results. Set Δn (x) = 1 − x2 /n + 1/n2 . (i) (ii) (iii) (iv) (v)

78

Chapter 3. Looking for New Moduli

Theorem 3.3.4 (Ivanov, [163] and [166]). Assume that w satisfies (3.16) with C(λ) = O(λc ) (λ → ∞) for some c > 0. (i) For every k ≥ 0 and for every f with f (k) ∈ Lp [−1, 1] we have En+k (f, w)p ≤ C(k)En (f (k) , w(Δn )k )p and En+k (f, w)p ≤ C(k) τ1 (f (k) , w(Δn )k , Δn )p where En (f, w)p =

inf (f − p)w1,p . p ∈ Pn

In particular, En+k (f )p ≤ C(k)En (f (k) , (Δn )k )p and En+k (f )p ≤ C(k) τ1 (f (k) , (Δn )k , Δn )p . (ii) If for each Q ∈ Πm , m ≤ n, wQ(k) (nΔn )k p ≤ C(k) mk wQp ,

(3.17)

then for every r ∈ [1, p] and f ∈ Lp [−1, 1], C(k)

≤ (j + 1)k−1 Es (f, w)p . nk j=0 n

τk (f, w, Δn )r,p

Using Koniagin [204] results we obtain Corollary 3.3.5. If f ∈ Lp [−1, 1] and r ∈ [1, p] and m ∈ N0 , then C(k, m)

(j + 1)k−1 Es (f, (nΔn )m )p . nk j=0 n

τk (f, (nΔn )m , Δn )r,p ≤ In particular

C(k)

(j + 1)k−1 Es (f )p . nk j=0 n

τk (f, Δn )r,p ≤

Corollary 3.3.6. If f ∈ Lp [−1, 1], r ∈ [1, p] and 0 < α < 1, one has En (f )p = O(n−α )

⇐⇒

τk (f, Δ(d))r,p = O(dα ).

The direct estimate in Theorem 3.3.4 is given in terms of the first τ modulus of the derivative f (k) . In [169] Ivanov presented the estimate in terms of the τ modulus of order k.

3.3. The τ modulus of Ivanov

79

Fix s > 0 and let us denote for W (s) the class of all weights w ∈ C[−1, 1] that have the following properties: for each x, t ∈ [−1, 1] with | x − t |≤ λΔn (x), 0 < w(x) ≤ C(λ) w(t) and, in the case λ ≥ 1,

0 < w(x) ≤ Cλs , w(t).

Theorem 3.3.7. Suppose that s > 0 and w ∈ W (s). If k ∈ N and f ∈ Lp [−1, 1] (1 ≤ p ≤ ∞), then En+k (f, w)p ≤ C(s, k) τk (f, w, Δn )p . Theorem 3.3.8. Suppose that s > 0, w ∈ W (s) and (3.17) holds. For f ∈ Lp [−1, 1] (1 ≤ p ≤ ∞), 0 < α < k, the following assertions are equivalent: (i) En (f, w)p = O(n−α ), (ii) τk (f, w, Δn )1,p = O(n−α ). In [173] Ivanov defined the τ modulus in a slightly different form: τk (f, w, ψ(t))r, p, [a, b] = w(·)ωk (f, ·, ψ(t, ·))r p,[a,b] where

 ωk (f, x, ψ(t, x))r =

and

1 2ψ(t, x)



ψ(t,x)

 k  Δ f (x)r dv

−ψ(t,x)

(3.18)

1/r

v

 # $ ωk (f, x, ψ(t, x))∞ = sup Δkh f (x) : | h |≤ ψ(t, x) .

Let us consider some types of weights. Two functions, v (continuous, strictly monotone, and v(0) = 0) and u, are associated with the weight w in neighborhoods of the end-points a and b. Let a and b be finite. Consider a neighborhood [a, d] of a or [d, b] of b; we write v(x) = x/w(a + x) for x ∈ (0, d − a] or v(x) = x/w(b − x) for ∈ (0, b − d], respectively. u is the inverse function to v, i.e., u(v(x)) = v(u(x)) = x. For a = 0 the functions u and w are connected by u(x) = v(u(x)w(v(x)) = xw(u(x)). Now consider the following classes: Type 1. w is non-decreasing, v is strictly increasing in [0, d], and v(0) = limx→0 v(x) = 0. For 0 < t ≤ v(d) we set ψ(t, x) = tw(x + u(t)).

(3.19)

Type 2. w is non-increasing and unbounded in (0, d] and, for every x ∈ (0, d/2], satisfies the inequality w(x) ≤ A2 w(2x). In this case ψ is also defined by (3.19).

80

Chapter 3. Looking for New Moduli

Type 3. v is non-increasing in (0, d] and, for every x ∈ (0, d], w satisfies the inequality (t0 = v(d)/2) w(x) ≤ Aw (x − t0 w(x)). In this case we define ψ(t, x) = tw(x). The weight w will satisfy the following global condition: There exist A5 ≥ 1, a < d3 < d1 < d2 < d4 < b, and weights w1 in [a, d1 ] and w2 in [d2 , b] of some of the types described above such that 1/A5 ≤ w(x)/w1 (x) ≤ A5 for x ∈ [a, d1 ], 1/A5 ≤ w(x)/w2 (x) ≤ A5 for x ∈ [d2 , b], and 1/A5 ≤ w(x) ≤ A5 for x ∈ [d3 , d4 ]. Sometimes we shall require v to satisfy the additional conditions  x dy v k (y) ≤ A1 v k (x), x ∈ (0, d] y 0

(3.20)

(3.21)

and sometimes we shall require w to satisfy the additional conditions u(λx) ≤ C(λ)u(x),

for any x > 0, λ ≥ 1,

λx ≤ d.

(3.22)

With vj , uj and ψj we denote the functions associated with the weight wj , j = 1, 2. Then we set ⎧ −1 x ∈ [a, d1 ], ⎪ ⎨ (2k) ψ1 (t, x), −1 (3.23) ψ(x, t) = (2k) ψ2 (t, x), x ∈ [d2 , b], ⎪ ⎩ linear and continuous, x ∈ [d1 , d2 ]. We also need the following condition: There is A > 1 such that 1/A ≤ ψ(t, x)/ψ(t, x) ≤ A for every x ∈ [a, b] and the weights w1 or w2 from (3.20) satisfy (3.21) and (3.22) provided they are of Type 1.

(3.24)

Theorem 3.3.9 (Ivanov, [173]). Let w satisfy (3.20) in [a, b] and let ψ satisfy (3.24) for 0 < t ≤ C(w). Then for every f ∈ Lp [a, b] + Wpk (w) we have C1 (k, w)τk (f, ψ(t))p,p ≤ K(f, tk , Lp , Wpk (w)) ≤ C1 (k, w)τk (f, ψ(t))1,p .  Let w(x) = x(1 − x), x ∈ √ [0, 1]. We can choose √ d1 = 1/3, d2 = 2/3, d3 = 1/4 and d√ x, and w2 (x) = 1 − x. Then u1 (t) = t2 4 = 3/4, w1 (x) = 2 2 and ψ1 (t, x) = t x + t . Therefore we can choose ψ(t, x) = tw(x) + t . Thus for ϕ(x) = x(1 − x) the last theorem yields C1 (k, w)τk (f, ψ(t))p,p ≤ K(f, tk , Lp , Wpk (ϕ)) ≤ C1 (k, w)τk (f, ψ(t))1,p . Let w be symmetry in [0, 1] (i.e., w(1 − x) = w(x)) and let w1 from (3.20) be of Type 1 in [0, 1/3] satisfying (3.21). Let us denote by u the function u1 ,

3.4. Ditzian-Totik moduli

81

corresponding to w1 . We set

 K ∗ (f, tk , Lp , Wpk (w)) = inf f − gp + tk wk g (k) p + u(t)k g (k) p .

Theorem 3.3.10 (Ivanov, [173]). Under the above assumption we have K(f, tk , Lp , Wpk (w)) ∼ K ∗ (f, tk , Lp [0, 1], Wpk (w)). Results related with characterizations of the best approximation of the best approximation by algebraic polynomials in terms of the τ -modulus were given in [163], [164], [165], [167], [168], [169], [171], [170], and [172]. The extension to Lp [−1, 1] with 0 < p < 1 was given by Tachev in [367] and [368].

3.4 Ditzian-Totik moduli In 1987 Ditzian and Totik published the book [102] where the following modulus was studied in detail. Let Δrh f (x) be the symmetric difference of order r (the difference is zero if some of the points are outside of the interval). For 1 ≤ p ≤ ∞ and f ∈ Lp [−1, 1] define ωϕr (f, t)p = sup Δhϕ f p , √

0 r, En (f )p ≤ C1 ωϕr (f, 1/n)p and for 0 < t < 1, ωϕr (f, t)p ≤ C2 tr



(n + 1)r−1 En (f )p .

0≤n≤1/t

Proof. Fix f . Taking into account Theorem 3.4.1, for each n we can find a function gn such that ∗ f − gn p + n−r ϕr gn(r) p + n−2r gn(r) p ≤ 2Kr,ϕ (f, tr )p ≤ Cωϕr (f, 1/n)p .

Thus, it is sufficient to find a good approximant for gn . First, from Proposition 3.4.2 with λ = r − 1 and g = g (r−1) , we obtain a polynomial Pn,1 such that  1 1 r−1 (r−1) r (r) r (r) (r) − Pn,1 )p ≤ C (Δn ) g p ≤ C1 ϕ g p + 2r g p (Δn ) (g nr n  1 ∗ f, r . ≤ C2 Kr,ϕ n p

3.4. Ditzian-Totik moduli

83

we apply the same proposition (with λ = r − 2) to the function g(u) = Now u (r−1) ((P (t)dt to obtain a polynomial Pn,2 ∈ Pn+1 for which n,1 (t) − g 0 (Δn )r−2 (g (r−2) − Pn,2 )p ≤ C (Δn )r−1 [g (r−1) − Pn,1 ]p  1 ∗ ≤ C3 Kr,ϕ f, r . n p Therefore, we can find a polynomial Pn,r ∈ Pn+r−1 such that  1 ∗ f, r gn − Pn,r p ≤ Cr Kr,ϕ . n p Since f − Pn,r p ≤ f − gn p + gn − Pn,r p , we have the direct result. For the converse results we use the Bernstein arguments, but now we use the Potapov inequality in Theorem 2.7.4. For t ∈ (0, |), let l = max{k 2k ≤ t } and {Pn } be the sequence of polynomials of the best approximation to f . From Theorems 3.4.1 and 2.7.4 one has   (r) ωϕr (f, t) ≤ CKr,ϕ (f, tr )p ≤ C f − P2l p + tr ϕr P2l p   l−1

= C f − P2l p + tr  ϕr (P2k+1 − P2k )(r) p k=0

 ≤ C1

E2l (f )p + t

≤ C2 tr



r

l−1

2

(k+1)r

 E2k (f )p

k=0

(n + 1)r−1 En (f )p .



0≤n≤1/t

In particular, from the last result we obtain the following characterization. Corollary 3.4.4. For 0 < α < r and f ∈ Lp [−1, 1] the following assertions are equivalent: (i) En (f )p ≤ Cn−α . (ii) ωϕr (f, t)p ≤ Ctα . The book contains different assertions concerning algebraic polynomials. The next result can be seen as an extension of an inequality due to Nikolskii and Stechkin. Theorem 3.4.5. Fix f ∈ Lp [−1, 1] and r ∈ N. Let Pn the best nth degree polynomial approximation to f in Lp [−1, 1], then ϕr Pn(r) p ≤ M ωϕr (f, 1/n)p ,

where ϕ(x) =

√ 1 − x2 and M is independent of f and n.

84

Chapter 3. Looking for New Moduli

For the converse they proved the following theorem. (r)

Theorem 3.4.6. Suppose that ϕr Pn p ≤ M nr ψ(1/n), where Pn is as in the last theorem and ψ(t) → 0 and t → 0. Then  En (f ) ≤ M

1/n

0

φ(t) dt t

 ωϕr (f, t)p

and

≤ M

0

t

φ(t) dt. t

Corollary 3.4.7. For 0 < α ≤ r and f ∈ Lp [−1, 1] the following assertions are equivalent: (r)

(i) ϕr Pn p ≤ Cnr−α . (ii) ωϕr (f, 1/n)p ≤ Cn−α . DeVore, Leviatan and Yu [90] extended the direct results in terms of the Ditzian-Totik modulus to Lp spaces, with 0 < p < 1. Ditzian, Jiang and Leviatan proved the converse results [101]. For these last spaces the methods based on a K-functional do not work, as was shown by Ditzian, Hristov and Ivanov [98]. In 2008, Dai, Ditzian and Tikhonov extended (1.10) to the case of algebraic approximation. Theorem 3.4.8 (Dai, Ditzian and Tikhonov, [80]). For 1 < p < ∞, s = max{p, 2} and f ∈ Lp [−1, 1], one has ⎛ tr ⎝



⎞1/s k sr−1 Ek (f )sp ⎠

≤ C(r)ωϕr (f, t)p

r≤k≤1/t

and

 t

r t

1/2

ωϕr+1 (f, u)sp du urs+1

1/s + tr Er (f )p ≤ C(r)ωϕr (f, t)p .

For the best approximation in C[−1, 1], the Timan-type results are pointwise and Ditzian-Totik are in norm. Ditzian and Jiang presented a possible way to unify both theories. √ Theorem 3.4.9 (Ditzian and Jiang, [99]). For λ ∈ [0, 1], ϕ(x) = 1 − x2 , there exists a constant C(r, λ) such that, for all f ∈ C[−1, 1] there exists a sequence {Pn } of polynomials such that,   1−λ  1 1 r ϕ(x) + . (3.25) | f (x) − Pn (f, x) | ≤ C(r, λ) ωϕλ f, n n If λ = 0, then we obtain the estimate in terms of the usual modulus of continuity and when λ = 1, we get the Ditzian-Totik estimate in norm. For the converse result Ditzian and Jiang proved the following. A similar result is not true for Lp spaces (1 ≤ p < ∞) and 0 ≤ λ < 1 (see [89] and [255]).

3.4. Ditzian-Totik moduli

85

Theorem 3.4.10 (Ditzian-Jiang, [99]). Fix s > 0 and let w be an increasing function satisfying w(μt) ≤ C(μs + 1)w(t). (3.26) If f ∈ C[−1, 1] and there exists a sequence {Pn } of polynomials such that   (3.27) | f (x) − Pn (x) |≤ Cw n−1 w(δn1−λ (x)) , then ωϕr λ (f, t) ≤ M tr



nr−1 w(n−1 ).

0 −2/p and v± (y) ∼ 1 on every interval [δ, 2], δ > 0, (c) for every ε > 0, y ε v± (y) are increasing and y −ε v± (y) are decreasing on (0, δ(ε)) for some δ(ε) > 0, and (d) for p = ∞ we may have γ1 = 0 or γ2 = 0 in which case v− (y) or v+ (y) have to be non-decreasing for small y. For f ∈ Lp [−1, 1] the main-part modulus is defined by Ωrϕ (f, t)w,p = sup wΔrhϕ f p,[−1+2r2h2 ,1−2r2 h2 ] . 0 (m − k + 1)/(m − k + r). √ Notice that Γnrmk (x) ≤ 1 − x2 /n for any 0 ≤ k ≤ m + 2 − r and for all x ∈ [−1, −1 + n−2 ] ∪ [1 − n−2 , 1]. The inequalities in the last theorem hold for all 0 ≤ k ≤ m, while Theorem 5.6.5 may not be true if k > m + 2 − r. It is also of interest to consider the special case r = 1 in the corollary. Corollary 3.4.26. For any n ≥ 2m+1 there exists a linear operator Pn : C m [−1,1] → Pn such that for every 0 ≤ k ≤ m, x ∈ [−1, 1] a function f ∈ C m [−1, 1], the following inequality holds: | f (k) (x) − Pn(k) (f, x) |≤ C(m)Γn (x)m−k ω(f (m) , Γn (x)), √ min{1 − x2 , 1 − x2 /n}. Moreover, Γn (x) cannot be replaced by where Γn (x) = √ min{(1 − x2 )α , 1 − x2 /n} with α > 1.

92

Chapter 3. Looking for New Moduli

By using the arguments of Leviatan, Kopotun also obtained an estimate in terms of the best approximation which improved those given by Kilgore in (2.40). Corollary 3.4.27. Then for any n ≥ 2m+1 and f ∈ C m [−1, 1] there is a polynomial Pn ∈ Pn such that for every 0 ≤ k ≤ m and x ∈ [−1, 1], m−k √ 1 − x2 |≤ C(r) min 1 − x , En−m (f (m) ). n 

|f

(k)

(x) −

Pn(k) (x)



2

3.4.5 A Banach space approach In [48] Butzer, Jansche and Stens considered the problem of generalizing the ideas of Butzer and Scherer ([51] and [52]) in such a way that we can also obtain the results of Ditzian and Totik. Solving the problem is justified by reasons of economy: to avoid many tricky and technical arguments. The main idea was to use Jackson-type inequalities and K-functionals with respect to a family of seminorms instead of a single seminorm. In particular they proved the Lipschitz spaces associated with Ditzian-Totik moduli coincide with the ones obtained by means of the Jacobi transform. For γ > 0, let Φ(γ) be the class of all functions φ : (0, 1] → R such that 0 < φ(s) ≤ φ(t) ≤ φ(1) < ∞, for 0 < s < t ≤ 1, limt→0+ φ(t) = 0 and  t

1

φ(u) du = O u1+γ



φ(t) tγ

.

Let us present the general theorem. We use the following notation: K(f, t, X, Y ) = inf {f − gX + t | g |Y } g∈Y and K ∗ (f, t, X, Y ) =

sup inf 0 0, M > 0 and n0 ∈ N. Let X be a normed space with norm  · X and Y ⊂ X a linear subspace with a seminorm | · |Y . Let {| · |X(t) }t∈(0,1] be a family of seminorms on X satisfying | f |X(t) ≤ | f |X(s) ≤ M f X ,

(0 < s ≤ t ≤ 1),

(3.28)

for a constant M , independent of f , s and t, and if {fn } is a Cauchy sequence in X with limn→∞ | fn |X(t) = 0 for all t ∈ (0, 1], then lim fn X = 0.

n→∞

(3.29)

3.4. Ditzian-Totik moduli

93

Let {Mn }∞ n=0 be a sequence of linear manifolds in X such that Mn ⊂ Mn+1 ⊂ Y,

(n ∈ N0 ),

| gn |Y ≤ M n gn X ,

(gn ∈ Mn ,

n ∈ N0 ),

gn X ≤ M | gn |X(2/n) ,

(gn ∈ Mn ,

n ≥ n0 ),

(3.30)

(f ∈ Y,

n ≥ n0 ),

(3.31)

γ

and En (f, X(1/n)) ≤ M n−γ | f |Y , where En (f, X(1/n)) =

inf | f − g |X(1/n) . g ∈ Mn

(a) For φ ∈ Φ(γ) and f ∈ X the following assertions are equivalent: inf f − gX = O(φ(1/n)), g ∈ Mn (ii) K(f, tγ , X, Y ) = O(φ(t)), (t → 0). (i) En (f ) =

(b) If



t 0

(n → ∞).

φ(u) du = O (φ(t)) , u

(3.32)

then (i) and (ii) are further equivalent to (ii)∗ K ∗ (f, tγ , X, Y ) = O (φ(t)),

(n → ∞).

(c) Assume that (3.32) holds. If for each f ∈ X and n ∈ N0 , there exists gn (f ) ∈ Mn such that En (f ) = f − gn (f ) and limn→∞ En (f ) = 0, then the assertions given above are equivalent to (iii) | gn (f ) |Y = O(nγ φ(1/n)),

(n → ∞).

(d) Fix δ > 0 and assume that the conditions in (a) hold. Let Z ⊂ X (Mn ⊂ Z) be a subspace with a seminorm | · |Z such that Z is a Banach space under the norm  · Z =  · X + | · |Z , En (f, X(1/n)) ≤ M n−δ | f |Z ,

(f ∈ Z,

n ≥ n0 ),

and | gn |Z ≤ M nδ gn X , If

 0

t

φ(u) du = O u1+δ

(gn ∈ Mn , 

φ(t) tδ

n ∈ N0 ).

,

(3.33)

then the assertion (i) is equivalent to (iv) f ∈ Z, f ∈ Z,

| f − gn (f ) |Z = O(nδ φ(1/n)), (n → ∞), En (f, Z) = inf | f − g |Z = O(nδ φ(1/n)), g ∈ Mn

Finally if φ ∈ Φ(γ) all the assertions given above are equivalent.

(t → 0).

94

Chapter 3. Looking for New Moduli

We only present a proof for (a), (b) and (c). Proof. (a) ((i) =⇒ (ii)). Fix t ∈ (0, 1] and k ∈ N0 such that 2−k−1 < t ≤ 2−k . In order to simplify, we assume that each Mn is an existence set. That is, for each n ∈ N, there exists Pn ∈ Mn such that En (f ) = f − Pn X . From the Bernstein-type inequality (| gn |Y ≤ M nγ gn X ), we know that     k k



  | P2k |Y = P1 + (P2j − P2j−1 ) ≤ C1 P1 X + M 2jγ (P2j − P2j−1 )X   j=1 j=1 Y ⎞ ⎞ ⎛ ⎛ k k



≤ C2 ⎝f X + 2jγ E2j (f )⎠ ≤ C3 ⎝f X + 2jγ φ(2−j )⎠ . j=0

j=0

Taking into account that Mn ⊂ Y and tγ

k 

2jγ φ(2−j ) ≤ C4 φ(t) (for φ ∈

j=0

Φ(γ)), one has K(f, tγ , X, Y ) ≤ f − P2k X + tγ | P2k |Y ⎛ ≤ C5 ⎝φ(2−k ) + tγ f X + tγ

k

⎞ 2jγ φ(2−j )⎠ ≤ C6 φ(t).

j=0

((ii) =⇒ (i)) Let us first verify a Jackson-type inequality. Fix g ∈ Y and ε > 0. For n ≥ n0 , take elements Qn2k ∈ M2k n such that | Q2k n − g |X(2−k n−1 ) ≤ E2k n (g, X(2−k n−1 )) + ε/2k . From (3.28) and (3.30), we know that   Q2k+1 n − Q2k n X ≤ M | Q2k+1 n − g |X(2−k−1 n−1 ) + | g − Q2k n |X(2−k n−1 )   ≤ M E2k+1 n (g,X(2−k−1 n−1 )) + E2k n (g,X(2−k n−1 )) + ε2−k . Therefore ∞

 Q2k+1 n − Q2k n X ≤ M

k=0

∞

 E2k n (g, X(2

−k −1

n

)) + ε .

k=0

Thus {Q2k n } is a Cauchy sequence in X and, for t ∈ (0, 1], lim | g − Q2k n |X(t) ≤ lim | g − Q2k n |X(2−k n−1 )

k→∞

k→∞

≤ lim M (E2k n (g, X(2−k n−1 )) + ε2−k ) = 0. k→∞

From (3.29) one has lim g − Q2k n X = 0 and there holds the representation k→∞

g − gn =

∞

(g2k+1 n − g2k n ),

k=0

where the convergence is considered with respect to the norm of X.

3.4. Ditzian-Totik moduli

95

Now En (g) ≤ g − gn X ≤

∞

 g2k+1 n − g2k n X ≤ M

ε+

k=0

∞

 E2k n (g, X(2

−k −1

n

)) .

k=0

Since ε > 0 is arbitrary we obtain (see (3.31)) En (g) ≤ M

∞

E2k n (g, X(2−k n−1 )) ≤ C1

k=0

∞

2−kγ n−γ | g |Y ≤ C2 n−γ | g |Y .

k=0

With this Jackson-type inequality (i) follows easily, since En (f ) ≤ En (f − g) + En (g) ≤ f − gX + C2 n−γ | g |Y and g ∈ Y is arbitrary. (b) (ii) =⇒ (ii)∗ . It follows from the inequality K ∗ (f, tγ , X, Y ) ≤ K(f, tγ , X, Y ). (ii)∗ =⇒ (i). Fix any g ∈ Y . For n ≥ n0 , En (f, X(1/n)) ≤ En (f − g, X(1/n)) + En (g, X(1/n))   ≤ M | f − g |X(n−1 ) +n−γ | g |Y . Since g ∈ Y is arbitrary, one has En (f ) ≤ M K ∗ (f, n−γ , X, Y ) ≤ Cφ(n−1 ). (c) (i) =⇒ (iii). Fix n ∈ N and k ∈ N0 such that 2k ≤ n < 2k+1 . Set gn = gn (f ). Taking into account the Bernstein-type inequality, (i) and the properties of φ, we obtain   k

   | gn |Y = g1 + (g2j − g2j−1 ) + (gn − g2k ) 

j=1

≤ C1 g1 X +

k

2 g2j − g2j−1 X + n gn − g2k X jγ

γ

j=1



≤ C2 f X + f − g1 X +

k

2 f − g2j X + n gn − f X + 2 jγ

γ

j=0



≤ C3 f X + n En (f ) + γ

k

≤ C4 f X + n φ(n γ

−1

)+

f − g2k X

2 E2j (f ) . jγ

j=0



(k+1)γ

k

j=0

jγ

2 φ(2

−j

) ≤ C3 nγ φ(n−1 ).

96

Chapter 3. Looking for New Moduli

(iii) =⇒ (i). From the Jackson-type inequality we know that, for n ≥ n0 and k ∈ N0 E2k n (f ) ≤ E2k n (f − g2k+1 n ) + E2k n (g2k+1 n ) ≤ f − g2k+1 n X + M (2k n)−γ | g2k+1 n |Y = E2k+1 n (f ) + M (2k n)−γ | g2k+1 n |Y . Since f − gn X → 0, condition (iii) yields En (f ) =

∞

(E2k n (f ) − E2k+1 n (f ))

k=0

≤M

∞

(2k n)−γ (2k+1 n)γ φ(2−k−1 n−1 ) ≤ Cφ(n−1 ).



k=0

Let us show how this result can be applied in weighted approximation. For (α,β) α, β > −1 and 1 ≤ p < ∞ let Lp [−1, 1] be the space of all f such that  f p,(α,β) =

1

−1

1/p | f (u) | wα,β (u)du

< ∞,

p

where wα,β (x) = (1 − x)α (1 + x)β . For α, β ≥ 0, Cα,β [−1, 1] is the space of all continuous functions, for which the limits limx→−1 wα,β (x)f (x) and limx→1 wα,β (x)f (x) exist, with the norm f ∞,(α,β) =

sup

| wα,β (x)f (x) | .

x∈[−1,1]

We need the family of seminorms used√in Theorem 3.4.28. In order to apply this theorem we fix c > 0 and, for t ∈ (0, 1/ c), set 

1/p

| f |X(t,c) =

| f (u) | wα,β (u)du p

,

f ∈ L(α,β) [−1, 1] p

I(t,c)

and | f |X(t,c) =

sup | wα,β (x)f (x) |,

f ∈ Cα,β [−1, 1],

x∈I(t,c)

√ where I(t, c) = [−1+ct2 , 1−ct2 ]. Moreover, for 1/ c ≤ t ≤ 1, we set | f |X(t,c) = 0. In the following we omit the interval [−1, 1] in the notation and X will be any one of the spaces defined above. √ Let us denote ϕ(x) = 1 − x2 and consider the differential operator (Ds f )(x) = ϕs (x)f (s) (x),

(x ∈ (−1, 1),

s ∈ N0 ).

(3.34)

3.4. Ditzian-Totik moduli

97

The associated Sobolev spaces are given by  s s , Wp,(α,β) = f ∈ L(α,β) [−1, 1] : f = F a.e., F ∈ ACloc , Ds F ∈ L(α,β) p p and $ # s W∞,(α,β) = f ∈ C s−1 [−1, 1] : f = F a.e., f ∈ C s−1 (−1, 1), Ds f ∈ Cα,β . s s → X is closed. Thus WX It can be proved that the linear operator Ds : WX is a Banach space with respect to the norm

f WXs , = f X + Ds f X ,

s (f ∈ WX ).

s Also in WX we consider the seminorm | f |WXs = Ds f X . In this case the Jacksontype inequality (3.31) can be proved in the form

En (g, X(1/n), c) ≤ M n−s Ds g,

s (g ∈ WX ),

(see Proposition 5.1 in [48], the proof follows some ideas of Ditzian and Totik [102]). The needed Bernstein-type inequality had been proved in 1974 by Khalilova [188]. For s ∈ N, Ds pn X ≤ M ns pn X ,

(pn ∈ Pn ,

n ∈ N0 ),

where the constant M is independent of n. Finally we need the inequality (3.30). But it can be proved √ that, for all c > 0 there exists a constant M > 0 such that, for all n ∈ N, n > 2c, pn X ≤ M | pn |X(1/n,c) ,

(pn ∈ Pn ).

A proof can be found in the Nevai book [268]. Another proof was given in the Ditzian and Totik book (Chapter 8.4 of [102]). Now we have all the necessary ingredients in order to use Theorem 3.4.28. But we first present a notion introduced by Ditzian and Totik. For s ∈ N and f ∈ X the weighted main-part modulus is defined by √ Ωs (f, t, X) = sup | Δhϕ(x) f (x) |X(h,2s2 ) , (0 < t < 1/( 2s)), 0